Admissible Bayes equivariant estimation of location vectors for spherically symmetric distributions with unknown scale

Yuzo Maruyama label=e1] maruyama@csis.u-tokyo.ac.jp [    William, E. Strawderman label=e2]straw@stat.rutgers.edu [ University of Tokyo\thanksmarkm1 and Rutgers University\thanksmarkm2
Abstract

This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form f(x,u)=η(p+n)/2f(η{xθ2+u2})𝑓𝑥𝑢superscript𝜂𝑝𝑛2𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2f(x,u)=\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\}), where η𝜂\eta is unknown. We show that the natural estimator x𝑥x is admissible for p=1,2𝑝12p=1,2. Also, for p3𝑝3p\geq 3, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form {1ξ(x/u)}x1𝜉𝑥norm𝑢𝑥\{1-\xi(x/\|u\|)\}x. In the Gaussian case, a variant of the James–Stein estimator, [1{(p2)/(n+2)}/{x2/u2+(p2)/(n+2)+1}]xdelimited-[]1𝑝2𝑛2superscriptnorm𝑥2superscriptnorm𝑢2𝑝2𝑛21𝑥[1-\{(p-2)/(n+2)\}/\{\|x\|^{2}/\|u\|^{2}+(p-2)/(n+2)+1\}]x, which dominates the natural estimator x𝑥x, is also admissible within this class. We also study the related regression model.

62C15,
62J07,
admissibility,
Stein’s phenomenon,
generalized Bayes,
Bayes equivariance,
keywords:
[class=AMS]
keywords:
\startlocaldefs\endlocaldefs

and

t1This work was partially supported by KAKENHI #25330035, #16K00040. t2This work was partially supported by grants from the Simons Foundation (#209035 and #418098 to William Strawderman).

1 Introduction

Let

(X,U)η(p+n)/2f(η{xθ2+u2})similar-to𝑋𝑈superscript𝜂𝑝𝑛2𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2(X,U)\sim\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\}) (1.1)

where Xp𝑋superscript𝑝X\in\mathbb{R}^{p} and Un𝑈superscript𝑛U\in\mathbb{R}^{n} and where θp𝜃superscript𝑝\theta\in\mathbb{R}^{p} and η+𝜂subscript\eta\in\mathbb{R}_{+} are unknown. We mainly assume

p3 and n2,𝑝3 and 𝑛2{\color[rgb]{0,0,0}p\geq 3\text{ and }n\geq 2}, (1.2)

and consider the problem of estimating θ𝜃\theta under scaled quadratic loss

L(δ,θ,η)=ηδθ2.𝐿𝛿𝜃𝜂𝜂superscriptnorm𝛿𝜃2L(\delta,\theta,\eta)=\eta\|\delta-\theta\|^{2}. (1.3)

In particular, we are interested in the admissibility among the class of equivariant estimators of the form

δξ(X,U)={1ξ(X/U)}X, where ξ:p.:subscript𝛿𝜉𝑋𝑈1𝜉𝑋norm𝑈𝑋 where 𝜉superscript𝑝\delta_{\xi}(X,U)=\left\{1-\xi(X/\|U\|)\right\}X,\text{ where }\xi:\mathbb{R}^{p}\to\mathbb{R}. (1.4)

We assume that f()0𝑓0f(\cdot)\geq 0 is defined so that each coordinate has variance 1/η1𝜂1/\eta. In particular, this implies that f()𝑓f(\cdot) in (1.1), satisfies

p+nf(v2)dv=1,p+nvi2f(v2)dv=1,formulae-sequencesubscriptsuperscript𝑝𝑛𝑓superscriptnorm𝑣2differential-d𝑣1subscriptsuperscript𝑝𝑛superscriptsubscript𝑣𝑖2𝑓superscriptnorm𝑣2differential-d𝑣1\int_{\mathbb{R}^{p+n}}f(\|v\|^{2})\mathrm{d}v=1,\ \int_{\mathbb{R}^{p+n}}v_{i}^{2}f(\|v\|^{2})\mathrm{d}v=1, (1.5)

for v=(v1,,vp+n)Tp+n𝑣superscriptsubscript𝑣1subscript𝑣𝑝𝑛Tsuperscript𝑝𝑛v=(v_{1},\dots,v_{p+n})^{\mathrm{\scriptscriptstyle T}}\in\mathbb{R}^{p+n}. Needless to say, this is a generalization of the Gaussian case where

fG(t)=1(2π)(p+n)/2exp(t/2)subscript𝑓𝐺𝑡1superscript2𝜋𝑝𝑛2𝑡2f_{G}(t)=\frac{1}{(2\pi)^{(p+n)/2}}\exp(-t/2)

and hence XNp(θ,η1I)similar-to𝑋subscript𝑁𝑝𝜃superscript𝜂1𝐼X\sim N_{p}(\theta,\eta^{-1}I) and U2η1χn2similar-tosuperscriptnorm𝑈2superscript𝜂1subscriptsuperscript𝜒2𝑛\|U\|^{2}\sim\eta^{-1}\chi^{2}_{n} are mutually independent.

In Section 2, we show that if an estimator of the form

δψ(X,U)={1ψ(X2/U2)}X, where ψ:+,:subscript𝛿𝜓𝑋𝑈1𝜓superscriptnorm𝑋2superscriptnorm𝑈2𝑋 where 𝜓subscript\delta_{\psi}(X,U)=\left\{1-\psi(\|X\|^{2}/\|U\|^{2})\right\}X,\text{ where }\psi:\mathbb{R}_{+}\to\mathbb{R}, (1.6)

is admissible within the class of all such estimators, then it is also admissible within a larger class of estimators, the class of all estimators of the form δξ(X,U)subscript𝛿𝜉𝑋𝑈\delta_{\xi}(X,U) given by (1.6). Note that the risk of an equivariant estimator of the form (1.6) is a function of λ=ηθ2𝜆𝜂superscriptnorm𝜃2\lambda=\eta\|\theta\|^{2}. Section 3 studies equivariant estimators of the form (1.6) which minimize the average risk with respect to a (proper) prior π(λ)𝜋𝜆\pi(\lambda) on the maximal invariant λ=ηθ2𝜆𝜂superscriptnorm𝜃2\lambda=\eta\|\theta\|^{2}. We give an expression for this average risk and for the equivariant estimator which effects the minimization. Additionally we show that this proper Bayes equivariant estimator is equivalent to the generalized (and not proper) Bayes estimator corresponding to the prior on (θ,η)𝜃𝜂(\theta,\eta)

π(θ,η)=η1{ηθ2}1p/2π(ηθ2).𝜋𝜃𝜂superscript𝜂1superscript𝜂superscriptnorm𝜃21𝑝2𝜋𝜂superscriptnorm𝜃2\pi(\theta,\eta)=\eta^{-1}\{\eta\|\theta\|^{2}\}^{1-p/2}\pi(\eta\|\theta\|^{2}).

Further we demonstrate that such an estimator is admissible among the class of estimators of the form (1.6) and hence (1.4).

Section 4, using Blyth’s (1951) method, extends the class of estimators which are admissible within the class of estimators of the form (1.6). One main result gives admissibility under π(λ)𝜋𝜆\pi(\lambda) including

π(λ)=λα for 1/2<α0,𝜋𝜆superscript𝜆𝛼 for 12𝛼0\pi(\lambda)=\lambda^{\alpha}\text{ for }-1/2<\alpha\leq 0,

for densities f𝑓f including the normal distribution and many generalized multivariate t𝑡t distributions. An interesting special case gives admissibility (within the class of equivariant estimators) of the generalized Bayes equivariant estimator corresponding to π(λ)1𝜋𝜆1\pi(\lambda)\equiv 1 or π(θ,η)=η1{ηθ2}1p/2𝜋𝜃𝜂superscript𝜂1superscript𝜂superscriptnorm𝜃21𝑝2\pi(\theta,\eta)=\eta^{-1}\{\eta\|\theta\|^{2}\}^{1-p/2}. Here the form of the generalized Bayes estimator is independent of the underlying density f()𝑓f(\cdot), as shown in Maruyama (2003). Further this estimator is minimax and dominates the James and Stein (1961) estimator

(1(p2)/(n+2)X2/U2)X1𝑝2𝑛2superscriptnorm𝑋2superscriptnorm𝑈2𝑋\left(1-\frac{(p-2)/(n+2)}{\|X\|^{2}/\|U\|^{2}}\right)X

provided f()𝑓f(\cdot) is non-increasing. Another interesting result is on a variant of the James–Stein estimator of the simple form

(1(p2)/(n+2)X2/U2+(p2)/(n+2)+1)X.1𝑝2𝑛2superscriptnorm𝑋2superscriptnorm𝑈2𝑝2𝑛21𝑋\left(1-\frac{(p-2)/(n+2)}{\|X\|^{2}/\|U\|^{2}+(p-2)/(n+2)+1}\right)X.

In the Gaussian case, this is the generalized Bayes equivariant estimator corresponding to

π(λ)=λp/2101(2πξ)p/2exp(λ2ξ)(ξ1+ξ)n/2dξ.𝜋𝜆superscript𝜆𝑝21superscriptsubscript01superscript2𝜋𝜉𝑝2𝜆2𝜉superscript𝜉1𝜉𝑛2differential-d𝜉\pi(\lambda)=\lambda^{p/2-1}\int_{0}^{\infty}\frac{1}{(2\pi\xi)^{p/2}}\exp\left(-\frac{\lambda}{2\xi}\right)\left(\frac{\xi}{1+\xi}\right)^{n/2}\mathrm{d}\xi.

It is admissible within the class of equivariant estimators, and is minimax.

In Section 5, we demonstrate that our setting (1.1) is regarded as a canonical form of a regression model with an intercept and a general spherically symmetric error distribution, where estimators of the form (1.6) corresponds to estimators of the vector of regression coefficients of the form {1ψ(R2)}β^1subscript𝜓superscript𝑅2^𝛽\{1-\psi_{\star}(R^{2})\}\hat{\beta} where ψ:(0,1):subscript𝜓01\psi_{\star}:(0,1)\to\mathbb{R}, β^^𝛽\hat{\beta} is the vector of least square estimators, and R2superscript𝑅2R^{2} is the coefficient of determination. Also estimators of the form (1.4) corresponds to estimators of the vector of regression coefficients of the form {1ξ(𝗍)}β^1subscript𝜉𝗍^𝛽\{1-\xi_{\star}(\mathsf{t})\}\hat{\beta} where ξ:p:subscript𝜉superscript𝑝\xi_{\star}:\mathbb{R}^{p}\to\mathbb{R} and 𝗍𝗍\mathsf{t} is the vector of t𝑡t values. Hence, from the regression viewpoint, these two classes of equivariant estimators are quite natural.

Section 6 gives some concluding remarks. Most of the proofs are given in a series of Appendices.

In this paper, we consider admissibility within the (restricted) class of equivariant estimators. The ultimate goal in this direction is clearly to show that some equivariant estimators of the form (1.6) are admissible among all estimators. Actually, when p=1,2𝑝12p=1,2, the natural estimator X𝑋X is admissible among all estimators, as shown in Appendix M. (This is not surprising but very expected. As far as we know, however, admissibility of X𝑋X with nuisance unknown scale η𝜂\eta, has not yet reported in any major journals and hence we provide the proof.) When the Stein phenomenon with p3𝑝3p\geq 3 occurs, considering general admissibility of equivariant estimators in the presence of nuisance parameter(s) has been a longstanding unsolved problem as mentioned in James and Stein (1961) and Brewster and Zidek (1974). While our results with p3𝑝3p\geq 3 do not resolve the general admissibility issue, they do advance substantially our understanding of admissibility within the class of equivariant estimators.

2 Admissibility in a broader sense

We consider two groups of transformations. In the following, let S=U2𝑆superscriptnorm𝑈2S=\|U\|^{2}.

Group I
XγΓX,θγΓθ,Sγ2S,ηη/γ2,formulae-sequence𝑋𝛾Γ𝑋formulae-sequence𝜃𝛾Γ𝜃formulae-sequence𝑆superscript𝛾2𝑆𝜂𝜂superscript𝛾2\displaystyle X\to\gamma\Gamma X,\quad\theta\to\gamma\Gamma\theta,\quad S\to\gamma^{2}S,\quad\eta\to\eta/\gamma^{2},

where Γ𝒪(p)Γ𝒪𝑝\Gamma\in\mathcal{O}(p), the group of p×p𝑝𝑝p\times p orthogonal matrices, and γ+𝛾subscript\gamma\in\mathbb{R}_{+}.

Group II
XγX,θγθ,Sγ2S,ηη/γ2,formulae-sequence𝑋𝛾𝑋formulae-sequence𝜃𝛾𝜃formulae-sequence𝑆superscript𝛾2𝑆𝜂𝜂superscript𝛾2\displaystyle X\to\gamma X,\quad\theta\to\gamma\theta,\quad S\to\gamma^{2}S,\quad\eta\to\eta/\gamma^{2},

where γ+𝛾subscript\gamma\in\mathbb{R}_{+}.

Equivariant estimators for Group I should satisfy

δ(γΓX,γ2S)=γΓδ(X,S),𝛿𝛾Γ𝑋superscript𝛾2𝑆𝛾Γ𝛿𝑋𝑆\displaystyle\delta(\gamma\Gamma X,\gamma^{2}S)=\gamma\Gamma\delta(X,S),

and reduce to estimators of the form

δψ={1ψ(X2/S)}Xsubscript𝛿𝜓1𝜓superscriptnorm𝑋2𝑆𝑋\delta_{\psi}=\left\{1-\psi(\|X\|^{2}/S)\right\}X (2.1)

where ψ:+:𝜓subscript\psi:\mathbb{R}_{+}\to\mathbb{R}. Equivariant estimator for Group II should satisfy

δ(γX,γ2S)=γδ(X,S),𝛿𝛾𝑋superscript𝛾2𝑆𝛾𝛿𝑋𝑆\displaystyle\delta(\gamma X,\gamma^{2}S)=\gamma\delta(X,S),

and reduce to estimator of the form

δξ={1ξ(X/S)}Xsubscript𝛿𝜉1𝜉𝑋𝑆𝑋\delta_{\xi}=\left\{1-\xi(X/\sqrt{S})\right\}X (2.2)

where ξ:p:𝜉superscript𝑝\xi:\mathbb{R}^{p}\to\mathbb{R}. It is useful to note the following.

Lemma 2.1.
  1. 1.

    The risk, R(θ,η,δψ)=E[ηδψθ2]𝑅𝜃𝜂subscript𝛿𝜓𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜓𝜃2R(\theta,\eta,\delta_{\psi})=E\left[\eta\|\delta_{\psi}-\theta\|^{2}\right], of an estimator δψsubscript𝛿𝜓\delta_{\psi}, is a function of ηθ2+𝜂superscriptnorm𝜃2subscript\eta\|\theta\|^{2}\in\mathbb{R}_{+}.

  2. 2.

    The risk, R(θ,η,δξ)𝑅𝜃𝜂subscript𝛿𝜉R(\theta,\eta,\delta_{\xi}), of an estimator δξsubscript𝛿𝜉\delta_{\xi}, is a function of η1/2θpsuperscript𝜂12𝜃superscript𝑝\eta^{1/2}\theta\in\mathbb{R}^{p}.

The standard proof is left to the reader.

Let two classes of estimators be

𝒟ψsubscript𝒟𝜓\displaystyle\mathcal{D}_{\psi} ={δψ with ψ:+ given by (2.1)},absentconditional-setsubscript𝛿𝜓 with 𝜓subscript given by (2.1)\displaystyle=\left\{\delta_{\psi}\text{ with }\psi:\mathbb{R}_{+}\to\mathbb{R}\text{ given by \eqref{eq:equiv.est.0}}\right\},
𝒟ξsubscript𝒟𝜉\displaystyle\mathcal{D}_{\xi} ={δξ with ξ:p given by (2.2)}.absentconditional-setsubscript𝛿𝜉 with 𝜉superscript𝑝 given by (2.2)\displaystyle=\left\{\delta_{\xi}\text{ with }\xi:\mathbb{R}^{p}\to\mathbb{R}\text{ given by \eqref{eq:equiv.est.1}}\right\}.

Clearly it follows that 𝒟ψ𝒟ξsubscript𝒟𝜓subscript𝒟𝜉\mathcal{D}_{\psi}\subset\mathcal{D}_{\xi}. We shall show that if δ𝒟ψ𝛿subscript𝒟𝜓\delta\in\mathcal{D}_{\psi} is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}, then it is admissible among the class 𝒟ξsubscript𝒟𝜉\mathcal{D}_{\xi}. The proof is due to Section 3 of Stein (1956), based on the compactness of the orthogonal group 𝒪(p)𝒪𝑝\mathcal{O}(p), and the continuity of, the problem.

Theorem 2.1.

If δ𝒟ψ𝛿subscript𝒟𝜓\delta\in\mathcal{D}_{\psi} is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}, then it is admissible among the class 𝒟ξsubscript𝒟𝜉\mathcal{D}_{\xi}.

Proof.

See Appendix A. ∎

In this paper, we will investigate admissibility among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}. Admissibility admissibility among the class 𝒟ξsubscript𝒟𝜉\mathcal{D}_{\xi} then follows by Theorem 2.1.

3 Proper Bayes equivariant estimators

Recall that an equivariant estimator for Group I is given by

δψ={1ψ(X2/S)}X.subscript𝛿𝜓1𝜓superscriptnorm𝑋2𝑆𝑋\delta_{\psi}=\left\{1-\psi(\|X\|^{2}/S)\right\}X. (3.1)

Since, as noted in Lemma 2.1, the risk function of the estimator δψ𝒟ψsubscript𝛿𝜓subscript𝒟𝜓\delta_{\psi}\in\mathcal{D}_{\psi}, R(θ,η,δψ)𝑅𝜃𝜂subscript𝛿𝜓R(\theta,\eta,\delta_{\psi}), depends only on ηθ2+𝜂superscriptnorm𝜃2subscript\eta\|\theta\|^{2}\in\mathbb{R}_{+}, it may be expressed as

R(θ,η,δψ)=R~(ηθ2,δψ).𝑅𝜃𝜂subscript𝛿𝜓~𝑅𝜂superscriptnorm𝜃2subscript𝛿𝜓R(\theta,\eta,\delta_{\psi})=\tilde{R}(\eta\|\theta\|^{2},\delta_{\psi}). (3.2)

Let λ=ηθ2+𝜆𝜂superscriptnorm𝜃2subscript\lambda=\eta\|\theta\|^{2}\in\mathbb{R}_{+}. We assume the prior density on λ𝜆\lambda is π(λ)𝜋𝜆\pi(\lambda), and in this section, we assume the propriety of π(λ)𝜋𝜆\pi(\lambda), that is,

0π(λ)dλ=1.superscriptsubscript0𝜋𝜆differential-d𝜆1\int_{0}^{\infty}\pi(\lambda)\mathrm{d}\lambda=1. (3.3)

For an equivariant estimator δψsubscript𝛿𝜓\delta_{\psi}, we define the Bayes equivariant risk as

B(δψ,π)=0R~(λ,δψ)π(λ)dλ.𝐵subscript𝛿𝜓𝜋superscriptsubscript0~𝑅𝜆subscript𝛿𝜓𝜋𝜆differential-d𝜆B(\delta_{\psi},\pi)=\int_{0}^{\infty}\tilde{R}(\lambda,\delta_{\psi})\pi(\lambda)\mathrm{d}\lambda. (3.4)

In this paper, the estimator δψsubscript𝛿𝜓\delta_{\psi} which minimizes B(δψ,π)𝐵subscript𝛿𝜓𝜋B(\delta_{\psi},\pi), is called the Bayes equivariant estimator and is denoted by δπsubscript𝛿𝜋\delta_{\pi}. In the following, let

cm=πm/2/Γ(m/2) for m+subscript𝑐𝑚superscript𝜋𝑚2Γ𝑚2 for 𝑚subscriptc_{m}=\pi^{m/2}/\Gamma(m/2)\text{ for }m\in\mathbb{N}_{+} (3.5)

and

π¯(λ)=cp1λ1p/2π(λ)¯𝜋𝜆superscriptsubscript𝑐𝑝1superscript𝜆1𝑝2𝜋𝜆\bar{\pi}(\lambda)=c_{p}^{-1}\lambda^{1-p/2}\pi(\lambda) (3.6)

so that π¯(μ2)¯𝜋superscriptnorm𝜇2\bar{\pi}(\|\mu\|^{2}) is a proper probability density on psuperscript𝑝\mathbb{R}^{p}, that is,

pπ¯(μ2)dμ=1.subscriptsuperscript𝑝¯𝜋superscriptnorm𝜇2differential-d𝜇1\int_{\mathbb{R}^{p}}\bar{\pi}(\|\mu\|^{2})\mathrm{d}\mu=1. (3.7)
Theorem 3.1.

Assume 0π(λ)dλ=1superscriptsubscript0𝜋𝜆differential-d𝜆1\int_{0}^{\infty}\pi(\lambda)\mathrm{d}\lambda=1 and that f𝑓f satisfies (1.5).

  1. 1.

    The Bayes equivariant risk B(δψ,π)𝐵subscript𝛿𝜓𝜋B(\delta_{\psi},\pi), (3.4), is given by

    B(δψ,π)=cnpψ(z2){ψ(z2)2(1zTM2(z,π)z2M1(z,π))}×z2M1(z,π)dz+p,𝐵subscript𝛿𝜓𝜋subscript𝑐𝑛subscriptsuperscript𝑝𝜓superscriptdelimited-∥∥𝑧2𝜓superscriptdelimited-∥∥𝑧221superscript𝑧Tsubscript𝑀2𝑧𝜋superscriptnorm𝑧2subscript𝑀1𝑧𝜋superscriptdelimited-∥∥𝑧2subscript𝑀1𝑧𝜋d𝑧𝑝\begin{split}B(\delta_{\psi},\pi)&=c_{n}\int_{\mathbb{R}^{p}}\psi(\|z\|^{2})\left\{\psi(\|z\|^{2})-2\left(1-\frac{z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)}{\|z\|^{2}M_{1}(z,\pi)}\right)\right\}\\ &\qquad\times\|z\|^{2}M_{1}(z,\pi)\mathrm{d}z+p,\end{split} (3.8)

    where cnsubscript𝑐𝑛c_{n} is given by (3.5) and

    M1(z,π)=η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη,M2(z,π)=θη(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη.formulae-sequencesubscript𝑀1𝑧𝜋double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂subscript𝑀2𝑧𝜋double-integral𝜃superscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂\begin{split}M_{1}(z,\pi)&=\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta,\\ M_{2}(z,\pi)&=\iint\theta\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta.\end{split} (3.9)
  2. 2.

    Given π(λ)𝜋𝜆\pi(\lambda), the minimizer of B(δψ,π)𝐵subscript𝛿𝜓𝜋B(\delta_{\psi},\pi) with respect to ψ𝜓\psi is

    ψπ(z2)=argminψB(δψ,π)=1zTM2(z,π)z2M1(z,π).subscript𝜓𝜋superscriptnorm𝑧2subscriptargmin𝜓𝐵subscript𝛿𝜓𝜋1superscript𝑧Tsubscript𝑀2𝑧𝜋superscriptnorm𝑧2subscript𝑀1𝑧𝜋\psi_{\pi}(\|z\|^{2})=\operatorname*{arg\,min}_{\psi}B(\delta_{\psi},\pi)=1-\frac{z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)}{\|z\|^{2}M_{1}(z,\pi)}. (3.10)
  3. 3.

    The Bayes equivariant estimator

    δπ={1ψπ(X2/S)}Xsubscript𝛿𝜋1subscript𝜓𝜋superscriptnorm𝑋2𝑆𝑋\displaystyle\delta_{\pi}=\left\{1-\psi_{\pi}(\|X\|^{2}/S)\right\}X (3.11)

    is equivalent to the generalized Bayes estimator of θ𝜃\theta with respect to the joint prior density η1ηp/2π¯(ηθ2)superscript𝜂1superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2\eta^{-1}\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2}) where π¯(λ)=cp1λ1p/2π(λ)¯𝜋𝜆superscriptsubscript𝑐𝑝1superscript𝜆1𝑝2𝜋𝜆\bar{\pi}(\lambda)=c_{p}^{-1}\lambda^{1-p/2}\pi(\lambda).

  4. 4.

    The Bayes equivariant estimator δπsubscript𝛿𝜋\delta_{\pi} is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}.

Proof.

See Appendix B. ∎

Remark 3.1.

As shown in Appendix C, the generalized Bayes estimator of θ𝜃\theta with respect to the joint prior density ηνηp/2π¯(ηθ2)superscript𝜂𝜈superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2\eta^{\nu}\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2}) for any ν𝜈\nu is a member of the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}. Part 3 of Theorem 3.1 applies only to the special case of ν=1𝜈1\nu=-1. The admissibility results of this section and of Section 4 apply only to this special case of ν=1𝜈1\nu=-1 and imply neither admissibility or inadmissibility of generalized Bayes estimators if ν1𝜈1\nu\neq-1. Also note that while π(λ)𝜋𝜆\pi(\lambda) is assumed proper in this section, the prior on (θ,η)𝜃𝜂(\theta,\eta), η1ηp/2π¯(ηθ2)superscript𝜂1superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2\eta^{-1}\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2}), is never proper since

0pη1ηp/2π¯(ηθ2)dθdη=0pη1π¯(μ2)dμdη=1×0dηη=.superscriptsubscript0subscriptsuperscript𝑝superscript𝜂1superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂superscriptsubscript0subscriptsuperscript𝑝superscript𝜂1¯𝜋superscriptnorm𝜇2differential-d𝜇differential-d𝜂1superscriptsubscript0d𝜂𝜂\displaystyle\int_{0}^{\infty}\!\!\int_{\mathbb{R}^{p}}\eta^{-1}\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta=\int_{0}^{\infty}\!\!\int_{\mathbb{R}^{p}}\eta^{-1}\bar{\pi}(\|\mu\|^{2})\mathrm{d}\mu\mathrm{d}\eta=1\times\int_{0}^{\infty}\frac{\mathrm{d}\eta}{\eta}=\infty.

4 Admissible Bayes equivariant estimators through the Blyth method

Even if π(λ)𝜋𝜆\pi(\lambda) on +subscript\mathbb{R}_{+} (and hence π¯(μ2)¯𝜋superscriptnorm𝜇2\bar{\pi}(\|\mu\|^{2}) on psuperscript𝑝\mathbb{R}^{p}) is improper, that is

pπ¯(μ2)dμ=0π(λ)dλ=,subscriptsuperscript𝑝¯𝜋superscriptnorm𝜇2differential-d𝜇superscriptsubscript0𝜋𝜆differential-d𝜆\int_{\mathbb{R}^{p}}\bar{\pi}(\|\mu\|^{2})\mathrm{d}\mu=\int_{0}^{\infty}\pi(\lambda)\mathrm{d}\lambda=\infty, (4.1)

the estimator δπsubscript𝛿𝜋\delta_{\pi} given by (3.11) can be defined if M1(z,π)subscript𝑀1𝑧𝜋M_{1}(z,\pi) and M2(z,π)subscript𝑀2𝑧𝜋M_{2}(z,\pi) given by (3.9) are both finite, and the admissibility of δπsubscript𝛿𝜋\delta_{\pi} within the class of equivariant estimators can be investigated through the Blyth method.

We consider the Bayes equivariant risk difference under πi(λ)subscript𝜋𝑖𝜆\pi_{i}(\lambda) which is proper, but not necessarily standardized; i.e., 0πi(λ)dλ<superscriptsubscript0subscript𝜋𝑖𝜆differential-d𝜆\int_{0}^{\infty}\pi_{i}(\lambda)\mathrm{d}\lambda<\infty. Let δπsubscript𝛿𝜋\delta_{\pi} and δπisubscript𝛿𝜋𝑖\delta_{\pi i} be Bayes equivariant estimators with respect to π(λ)𝜋𝜆\pi(\lambda) and πi(λ)subscript𝜋𝑖𝜆\pi_{i}(\lambda), respectively. By Parts 1 and 2 of Theorem 3.1, the Bayes equivariant risk difference under πi(λ)subscript𝜋𝑖𝜆\pi_{i}(\lambda) is given as follows:

diffB(δπ,δπi;πi)diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\displaystyle\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i}) (4.2)
=0{R(λ,δπ)R(λ,δπi)}πi(λ)dλabsentsuperscriptsubscript0𝑅𝜆subscript𝛿𝜋𝑅𝜆subscript𝛿𝜋𝑖subscript𝜋𝑖𝜆differential-d𝜆\displaystyle=\int_{0}^{\infty}\{R(\lambda,\delta_{\pi})-R(\lambda,\delta_{\pi i})\}\pi_{i}(\lambda)\mathrm{d}\lambda
=cnp({ψπ2(z2)2ψπ(z2)ψπi(z2)}\displaystyle=c_{n}\int_{\mathbb{R}^{p}}\left(\left\{\psi_{\pi}^{2}(\|z\|^{2})-2\psi_{\pi}(\|z\|^{2})\psi_{\pi i}(\|z\|^{2})\right\}\right.
{ψπi2(z2)2ψπi(z2)ψπi(z2)})z2M1(z,πi)dz\displaystyle\qquad\left.-\left\{\psi_{\pi i}^{2}(\|z\|^{2})-2\psi_{\pi i}(\|z\|^{2})\psi_{\pi i}(\|z\|^{2})\right\}\right)\|z\|^{2}M_{1}(z,\pi_{i})\mathrm{d}z
=cnpdiffB¯(z;δπ,δπi;πi)dz,absentsubscript𝑐𝑛subscriptsuperscript𝑝¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖differential-d𝑧\displaystyle=c_{n}\int_{\mathbb{R}^{p}}\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})\mathrm{d}z,

where cnsubscript𝑐𝑛c_{n} is given by (3.5) and where

diffB¯(z;δπ,δπi;πi)={ψπ(z2)ψπi(z2)}2z2M1(z,πi).¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖superscriptsubscript𝜓𝜋superscriptnorm𝑧2subscript𝜓𝜋𝑖superscriptnorm𝑧22superscriptnorm𝑧2subscript𝑀1𝑧subscript𝜋𝑖\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})=\{\psi_{\pi}(\|z\|^{2})-\psi_{\pi i}(\|z\|^{2})\}^{2}\|z\|^{2}M_{1}(z,\pi_{i}). (4.3)

There are several versions of the Blyth method. For our purpose, the following version from Brown (1971) and Brown and Hwang (1982) is useful.

Theorem 4.1.

Assume that the sequence πi(λ)subscript𝜋𝑖𝜆\pi_{i}(\lambda) for i=1,2,,𝑖12i=1,2,\dots, satisfies

  1. BL.1

    π1(λ)π2(λ)subscript𝜋1𝜆subscript𝜋2𝜆italic-…\pi_{1}(\lambda)\leq\pi_{2}(\lambda)\leq\dots for any λ0𝜆0\lambda\geq 0 and limiπi(λ)=π(λ)subscript𝑖subscript𝜋𝑖𝜆𝜋𝜆\lim_{i\to\infty}\pi_{i}(\lambda)=\pi(\lambda).

  2. BL.2

    0πi(λ)dλ<superscriptsubscript0subscript𝜋𝑖𝜆differential-d𝜆\displaystyle\int_{0}^{\infty}\pi_{i}(\lambda)\mathrm{d}\lambda<\infty for any fixed i𝑖i.

  3. BL.3

    01π1(λ)dλ>γsuperscriptsubscript01subscript𝜋1𝜆differential-d𝜆𝛾\displaystyle\int_{0}^{1}\pi_{1}(\lambda)\mathrm{d}\lambda>\gamma for some positive γ>0𝛾0\gamma>0.

  4. BL.4

    limidiffB(δπ,δπi;πi)=0subscript𝑖diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖0\displaystyle\lim_{i\to\infty}\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i})=0.

Then δπsubscript𝛿𝜋\delta_{\pi} is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}.

Proof.

See Appendix D. ∎

We consider the following assumptions on π𝜋\pi in addition to (4.1).

Assumptions on π𝜋\pi.

  1. A.1

    π(λ)𝜋𝜆\pi(\lambda) is differentiable.

  2. A.2

    (Behavior around the origin) For λ[0,1]𝜆01\lambda\in[0,1], there exist α>1/2𝛼12\alpha>-1/2 and ν(λ)𝜈𝜆\nu(\lambda) such that

    π(λ)=λαν(λ),𝜋𝜆superscript𝜆𝛼𝜈𝜆\displaystyle\pi(\lambda)=\lambda^{\alpha}\nu(\lambda),

    where

    0<ν(0)< and limλ0λν(λ)=0.0𝜈0 and subscript𝜆0𝜆superscript𝜈𝜆0\displaystyle 0<\nu(0)<\infty\text{ and }\lim_{\lambda\to 0}\lambda\nu^{\prime}(\lambda)=0.
  3. A.3

    (Asymptotic behavior) Let κ(λ)=λπ(λ)/π(λ)𝜅𝜆𝜆superscript𝜋𝜆𝜋𝜆\kappa(\lambda)=\lambda\pi^{\prime}(\lambda)/\pi(\lambda). Either A.3A.3.1 or A.3A.3.2 is assumed;

    1. A.3.1

      1limλκ(λ)<01subscript𝜆𝜅𝜆0\displaystyle-1\leq\lim_{\lambda\to\infty}\kappa(\lambda)<0

    2. A.3.2

      limλκ(λ)=0subscript𝜆𝜅𝜆0\displaystyle\lim_{\lambda\to\infty}\kappa(\lambda)=0. Further either A.3(A.3.2)A.3.2.1 or A.3(A.3.2)A.3.2.2 is assumed;

      1. A.3.2.1

        κ(λ)𝜅𝜆\kappa(\lambda) is eventually monotone increasing and approaches 00 from below.

      2. A.3.2.2

        lim supλ{logλ}|κ(λ)|<1subscriptlimit-supremum𝜆𝜆𝜅𝜆1\displaystyle\limsup_{\lambda\to\infty}\,\{\log\lambda\}|\kappa(\lambda)|<1.

A typical prior π(λ)𝜋𝜆\pi(\lambda) satisfying Assumptions A.1A.3, corresponding to a generalized Strawderman’s (1971) prior, is given by

π(λ;α,β,b)=cpλp/21b1(2πξ)p/2exp(λ2ξ)(ξb)α(1+ξ)βdξ.𝜋𝜆𝛼𝛽𝑏subscript𝑐𝑝superscript𝜆𝑝21superscriptsubscript𝑏1superscript2𝜋𝜉𝑝2𝜆2𝜉superscript𝜉𝑏𝛼superscript1𝜉𝛽differential-d𝜉\pi(\lambda;\alpha,\beta,b)=c_{p}\lambda^{p/2-1}\int_{b}^{\infty}\frac{1}{(2\pi\xi)^{p/2}}\exp\left(-\frac{\lambda}{2\xi}\right)(\xi-b)^{\alpha}(1+\xi)^{\beta}\mathrm{d}\xi. (4.4)

Assumptions A.1A.3 are satisfied when {1α+β0,α>1,b>0}formulae-sequence1𝛼𝛽0formulae-sequence𝛼1𝑏0\{-1\leq\alpha+\beta\leq 0,\ \alpha>-1,\ b>0\} or {1α+β0,α>1/2,b=0}formulae-sequence1𝛼𝛽0formulae-sequence𝛼12𝑏0\{-1\leq\alpha+\beta\leq 0,\alpha>-1/2,b=0\}. See Appendix L for the proof. Note that the power prior π(λ)=λα𝜋𝜆superscript𝜆𝛼\pi(\lambda)=\lambda^{\alpha} for /2<α0-/2<\alpha\leq 0, which will be considered in Section 4.1, corresponds to the case β=0𝛽0\beta=0 and b=0𝑏0b=0.

For a generalized prior π(λ)𝜋𝜆\pi(\lambda) satisfying Assumptions A.1A.3, consider the sequence given by πi(λ)=π(λ)hi2(λ)subscript𝜋𝑖𝜆𝜋𝜆superscriptsubscript𝑖2𝜆\pi_{i}(\lambda)=\pi(\lambda)h_{i}^{2}(\lambda) where hi(λ)subscript𝑖𝜆h_{i}(\lambda), for λ0𝜆0\lambda\geq 0 and i=1,2,,𝑖12i=1,2,\dots, is defined by

hi(λ)=1loglog(λ+e)loglog(λ+e+i),subscript𝑖𝜆1𝜆𝑒𝜆𝑒𝑖\displaystyle h_{i}(\lambda)=1-\frac{\log\log(\lambda+e)}{\log\log(\lambda+e+i)}, (4.5)

and e=exp(1)𝑒1e=\exp(1). It is clear that πisubscript𝜋𝑖\pi_{i} satisfies BL.1 of Theorem 4.1. In Lemma E.2 of Appendix E, we show that πisubscript𝜋𝑖\pi_{i} also satisfies BL.2 and BL.3 of Theorem 4.1.

For BL.4, note that diffB(δπ,δπi;πi)diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i}) given by (4.2) is a functional of f𝑓f as well as π𝜋\pi and πisubscript𝜋𝑖\pi_{i}. Some additional assumptions on f𝑓f (as well as (1.5)) are required as follows;

Assumptions on f𝑓f.

  1. F.1

    0<f(t)<0𝑓𝑡0<f(t)<\infty for any t0𝑡0t\geq 0.

  2. F.2

    f𝑓f is differentiable.

  3. F.3

    the asymptotic behavior: Either F.3F.3.1 or F.3F.3.2 is assumed;

    1. F.3.1

      lim supttf(t)f(t)<p+n22subscriptlimit-supremum𝑡𝑡superscript𝑓𝑡𝑓𝑡𝑝𝑛22\displaystyle\limsup_{t\to\infty}\,t\frac{f^{\prime}(t)}{f(t)}<-\frac{p+n}{2}-2.

    2. F.3.2

      lim supttf(t)f(t)<p+n23subscriptlimit-supremum𝑡𝑡superscript𝑓𝑡𝑓𝑡𝑝𝑛23\displaystyle\limsup_{t\to\infty}\,t\frac{f^{\prime}(t)}{f(t)}<-\frac{p+n}{2}-3.

We note that, in addition to the normal distribution,

fG(t)=(2π)(p+n)/2exp(t/2),subscript𝑓𝐺𝑡superscript2𝜋𝑝𝑛2𝑡2f_{G}(t)=(2\pi)^{-(p+n)/2}\exp(-t/2),

an interesting flatter tailed class, also satisfying Assumptions F.1F.3, is given by the multivariate generalized Student t𝑡t with

f(t;a,b)𝑓𝑡𝑎𝑏\displaystyle f(t;a,b) =0fG(t/g)g(p+n)/2ga/21Γ(a/2)(2/b)a/2exp(b2g)dgabsentsuperscriptsubscript0subscript𝑓𝐺𝑡𝑔superscript𝑔𝑝𝑛2superscript𝑔𝑎21Γ𝑎2superscript2𝑏𝑎2𝑏2𝑔differential-d𝑔\displaystyle=\int_{0}^{\infty}\frac{f_{G}(t/g)}{g^{(p+n)/2}}\frac{g^{-a/2-1}}{\Gamma(a/2)(2/b)^{a/2}}\exp\left(-\frac{b}{2g}\right)\mathrm{d}g
=Γ((p+n+a)/2)(πb)(p+n)/2Γ(a/2)(1+t/b)(p+n+a)/2.absentΓ𝑝𝑛𝑎2superscript𝜋𝑏𝑝𝑛2Γ𝑎2superscript1𝑡𝑏𝑝𝑛𝑎2\displaystyle=\frac{\Gamma((p+n+a)/2)}{(\pi b)^{(p+n)/2}\Gamma(a/2)}(1+t/b)^{-(p+n+a)/2}.

For Assumptions F.3F.3.1 and F.3F.3.2, a>4𝑎4a>4 and a>6𝑎6a>6 are needed respectively.

The main result on admissibility of δπsubscript𝛿𝜋\delta_{\pi} given by (3.11) among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi} through the Blyth method is as follows.

Theorem 4.2.

Case I

Assume Assumptions A.1, A.2 and A.3A.3.1 on π𝜋\pi and Assumptions F.1, F.2 and F.3F.3.1 on f𝑓f. Then the estimator δπsubscript𝛿𝜋\delta_{\pi} given by (3.11) is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}.

Case II

Assume Assumptions A.1, A.2 and A.3A.3.2 on π𝜋\pi and Assumptions F.1, F.2 and F.3F.3.2 on f𝑓f. Then the estimator δπsubscript𝛿𝜋\delta_{\pi} given by (3.11) is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}.

The proof of Theorem 4.2, or essentially, equivalently the proof of BL.4,

limidiffB(δπ,δπi;πi)=0subscript𝑖diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖0\lim_{i\to\infty}\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i})=0

under the above Assumptions, is provided in Appendices F, G and H. Prior to these sections, some preliminary needed results on π𝜋\pi, f𝑓f and πisubscript𝜋𝑖\pi_{i} are given in Appendix E.

Remark 4.1.

The basic idea behind the sequence hisubscript𝑖h_{i} given by (4.5) comes from the hisubscript𝑖h_{i} of Brown and Hwang (1982),

hi(λ)={1λ11logλ/logi1λi0λ>i.subscript𝑖𝜆cases1𝜆11𝜆𝑖1𝜆𝑖0𝜆𝑖\displaystyle h_{i}(\lambda)=\begin{cases}1&\lambda\leq 1\\ 1-\log\lambda/\log i&1\leq\lambda\leq i\\ 0&\lambda>i.\end{cases} (4.6)

A smoothed version of the above is

hi(λ)=1log(λ+1)log(λ+1+i).subscript𝑖𝜆1𝜆1𝜆1𝑖h_{i}(\lambda)=1-\frac{\log(\lambda+1)}{\log(\lambda+1+i)}. (4.7)

The sequence hisubscript𝑖h_{i} given by (4.5) is more slowly changing in both r𝑟r and i𝑖i, in order to handle priors with flatter tail than treated in Brown and Hwang (1982). Also, with smooth πi=πhi2subscript𝜋𝑖𝜋superscriptsubscript𝑖2\pi_{i}=\pi h_{i}^{2}, the proofs become simpler.

Remark 4.2.

Assumption A.3 is a sufficient condition for

1dλλπ(λ)=1dλλp/2π¯(λ)=,superscriptsubscript1d𝜆𝜆𝜋𝜆superscriptsubscript1d𝜆superscript𝜆𝑝2¯𝜋𝜆\int_{1}^{\infty}\frac{\mathrm{d}\lambda}{\lambda\pi(\lambda)}=\infty\ \Leftrightarrow\ \int_{1}^{\infty}\frac{\mathrm{d}\lambda}{\lambda^{p/2}\bar{\pi}(\lambda)}=\infty, (4.8)

which is related to admissibility in the known variance case as follows. Maruyama (2009) showed that, in the problem of estimating μ𝜇\mu of XNp(μ,I)similar-to𝑋subscript𝑁𝑝𝜇𝐼X\sim N_{p}(\mu,I), regularly varying priors g(μ2)𝑔superscriptnorm𝜇2g(\|\mu\|^{2}) with

1dλλp/2g(λ)=superscriptsubscript1d𝜆superscript𝜆𝑝2𝑔𝜆\int_{1}^{\infty}\frac{\mathrm{d}\lambda}{\lambda^{p/2}g(\lambda)}=\infty (4.9)

lead to admissibility, that is, the (generalized) Bayes estimator

X+logmg(X2)𝑋subscript𝑚𝑔superscriptnorm𝑋2X+\nabla\log m_{g}(\|X\|^{2})

where

mg(x2)=1(2π)p/2exp(xμ2/2)g(μ2)dμsubscript𝑚𝑔superscriptnorm𝑥21superscript2𝜋𝑝2superscriptnorm𝑥𝜇22𝑔superscriptnorm𝜇2differential-d𝜇m_{g}(\|x\|^{2})=\frac{1}{(2\pi)^{p/2}}\int\exp(-\|x-\mu\|^{2}/2)g(\|\mu\|^{2})\mathrm{d}\mu (4.10)

is admissible. As Maruyama (2009) pointed out, the sufficient condition (4.9), which depends directly on the prior g(μ2)𝑔superscriptnorm𝜇2g(\|\mu\|^{2}), is closely related to Brown’s (1971) sufficient condition for admissibility

1drrp/2mg(r)=,superscriptsubscript1d𝑟superscript𝑟𝑝2subscript𝑚𝑔𝑟\int_{1}^{\infty}\frac{\mathrm{d}r}{r^{p/2}m_{g}(r)}=\infty,

which depends on the marginal distribution and only indirectly on the prior. Note also that Assumption A.3 is tight for the non-integrability of (4.8), in the sense that, among the class π(λ){logλ}b𝜋𝜆superscript𝜆𝑏\pi(\lambda)\approx\{\log\lambda\}^{b} with b𝑏b\in\mathbb{R},

1dλλ{logλ}1ϵ=,1dλλlogλ=,1dλλ{logλ}1+ϵ<formulae-sequencesuperscriptsubscript1d𝜆𝜆superscript𝜆1italic-ϵformulae-sequencesuperscriptsubscript1d𝜆𝜆𝜆superscriptsubscript1d𝜆𝜆superscript𝜆1italic-ϵ\int_{1}^{\infty}\frac{\mathrm{d}\lambda}{\lambda\{\log\lambda\}^{1-\epsilon}}=\infty,\ \int_{1}^{\infty}\frac{\mathrm{d}\lambda}{\lambda\log\lambda}=\infty,\ \int_{1}^{\infty}\frac{\mathrm{d}\lambda}{\lambda\{\log\lambda\}^{1+\epsilon}}<\infty

where in the first expression {logλ}1ϵsuperscript𝜆1italic-ϵ\{\log\lambda\}^{1-\epsilon} satisfies Assumption A.3, and in the second, logλ𝜆\log\lambda does not satisfy Assumption A.3. Actually, in the third case, {logλ}1+ϵsuperscript𝜆1italic-ϵ\{\log\lambda\}^{1+\epsilon}, the corresponding Bayes equivariant estimator is inadmissible as shown in Maruyama and Strawderman (2017).

4.1 Some interesting cases

Here we present three interesting special cases of our main general theorem.

Corollary 4.1.

Assume Assumptions F.1, F.2 and F.3F.3.2 on f𝑓f.

  1. 1.

    Then δπsubscript𝛿𝜋\delta_{\pi} with π1𝜋1\pi\equiv 1, or equivalently the generalized Bayes estimator under the prior on (θ,η)𝜃𝜂(\theta,\eta) given by

    η1ηp/2{ηθ2}(2p)/2,superscript𝜂1superscript𝜂𝑝2superscript𝜂superscriptnorm𝜃22𝑝2\displaystyle\eta^{-1}\eta^{p/2}\left\{\eta\|\theta\|^{2}\right\}^{(2-p)/2},

    is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}.

  2. 2.

    The form of the generalized Bayes estimator does not depend on f𝑓f and is given by {1ψ0(W)}X1subscript𝜓0𝑊𝑋\left\{1-\psi_{0}(W)\right\}X where W=X2/S𝑊superscriptnorm𝑋2𝑆W=\|X\|^{2}/S and

    ψ0(w)=01tp/21(1+wt)(p+n)/21dt01tp/22(1+wt)(p+n)/21dt.subscript𝜓0𝑤superscriptsubscript01superscript𝑡𝑝21superscript1𝑤𝑡𝑝𝑛21differential-d𝑡superscriptsubscript01superscript𝑡𝑝22superscript1𝑤𝑡𝑝𝑛21differential-d𝑡\displaystyle\psi_{0}(w)=\frac{\int_{0}^{1}t^{p/2-1}(1+wt)^{-(p+n)/2-1}\mathrm{d}t}{\int_{0}^{1}t^{p/2-2}(1+wt)^{-(p+n)/2-1}\mathrm{d}t}.
  3. 3.

    This estimator is minimax simultaneously for all such f𝑓f.

  4. 4.

    This estimator dominates the James–Stein estimator

    (1p2n+2SX2)X1𝑝2𝑛2𝑆superscriptnorm𝑋2𝑋\displaystyle\left(1-\frac{p-2}{n+2}\frac{S}{\|X\|^{2}}\right)X

    if f𝑓f is nonincreasing.

Proof.

For Part 1, Assumptions A.1, A.2 and A.3A.3.2 are satisfied by π(λ)1𝜋𝜆1\pi(\lambda)\equiv 1. Parts 2 and 4 are both shown by Maruyama (2003). Part 3 is shown by Cellier, Fourdrinier and Robert (1989). ∎

Corollary 4.2.

Assume Assumptions F.1, F.2 and F.3F.3.1 on f𝑓f. Let α(1/2,0)𝛼120\alpha\in(-1/2,0).

  1. 1.

    Then δπsubscript𝛿𝜋\delta_{\pi} with π(λ)=λα𝜋𝜆superscript𝜆𝛼\pi(\lambda)=\lambda^{\alpha}, or equivalently the generalized Bayes estimator under the prior on (θ,η)𝜃𝜂(\theta,\eta) given by

    η1ηp/2{ηθ2}α+(2p)/2,superscript𝜂1superscript𝜂𝑝2superscript𝜂superscriptnorm𝜃2𝛼2𝑝2\displaystyle\eta^{-1}\eta^{p/2}\left\{\eta\|\theta\|^{2}\right\}^{\alpha+(2-p)/2},

    is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}.

  2. 2.

    The form of the estimator does not depend on f𝑓f and is given by {1ψα(W)}X1subscript𝜓𝛼𝑊𝑋\left\{1-\psi_{\alpha}(W)\right\}X where W=X2/S𝑊superscriptnorm𝑋2𝑆W=\|X\|^{2}/S and

    ψα(w)=01tp/2α1(1t)α(1+wt)(p+n)/21dt01tp/2α2(1t)α(1+wt)(p+n)/21dt.subscript𝜓𝛼𝑤superscriptsubscript01superscript𝑡𝑝2𝛼1superscript1𝑡𝛼superscript1𝑤𝑡𝑝𝑛21differential-d𝑡superscriptsubscript01superscript𝑡𝑝2𝛼2superscript1𝑡𝛼superscript1𝑤𝑡𝑝𝑛21differential-d𝑡\psi_{\alpha}(w)=\frac{\int_{0}^{1}t^{p/2-\alpha-1}(1-t)^{\alpha}(1+wt)^{-(p+n)/2-1}\mathrm{d}t}{\int_{0}^{1}t^{p/2-\alpha-2}(1-t)^{\alpha}(1+wt)^{-(p+n)/2-1}\mathrm{d}t}. (4.11)
  3. 3.

    This estimator is minimax when

    (5+2p2+3pn+2)1α<0.superscript52𝑝23𝑝𝑛21𝛼0-\left(5+\frac{2}{p-2}+\frac{3p}{n+2}\right)^{-1}\leq\alpha<0.
Proof.

For Part 1, Assumptions A.1, A.2 and A.3A.3.1 are satisfied by π(λ)=λα𝜋𝜆superscript𝜆𝛼\pi(\lambda)=\lambda^{\alpha} for α(1/2,0)𝛼120\alpha\in(-1/2,0). Part 2 is shown by Maruyama (2003). For Part 3, see Maruyama and Strawderman (2009) and Appendix J. ∎

The following corollary relates to the so-called “simple Bayes estimators” from Maruyama and Strawderman (2005).

Corollary 4.3.

Assume f𝑓f is Gaussian. Then the simple Bayes estimator

(1a(a+1)(b+1)+X2/S)X1𝑎𝑎1𝑏1superscriptnorm𝑋2𝑆𝑋\displaystyle\left(1-\frac{a}{(a+1)(b+1)+\|X\|^{2}/S}\right)X

with a(p2)/(n+2)𝑎𝑝2𝑛2a\geq(p-2)/(n+2) and b0𝑏0b\geq 0 is admissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}. Furthermore, the estimator with (p2)/(n+2)a2(p2)/(n+2)𝑝2𝑛2𝑎2𝑝2𝑛2(p-2)/(n+2)\leq a\leq 2(p-2)/(n+2) is minimax.

Proof.

The estimator is (generalized) Bayes equivariant estimator with respect to π(λ;α,β,b)𝜋𝜆𝛼𝛽𝑏\pi(\lambda;\alpha,\beta,b) given by (4.4) with β=n/2𝛽𝑛2\beta=-n/2 and α=(p+n)/{2(a+1)}1𝛼𝑝𝑛2𝑎11\alpha=(p+n)/\{2(a+1)\}-1. See Maruyama and Strawderman (2005) and Appendix K. ∎

5 Canonical form of the regression setup

Suppose a linear regression model is used to relate y𝑦y to the p𝑝p predictors z1,,zpsubscript𝑧1subscript𝑧𝑝z_{1},\dots,z_{p},

y=α1m+Zβ+η1/2ϵ𝑦𝛼subscript1𝑚𝑍𝛽superscript𝜂12italic-ϵy=\alpha 1_{m}+Z\beta+\eta^{-1/2}\epsilon (5.1)

where α𝛼\alpha is an unknown intercept parameter, 1msubscript1𝑚1_{m} is an m×1𝑚1m\times 1 vector of ones, Z=(z1,,zp)𝑍subscript𝑧1subscript𝑧𝑝Z=(z_{1},\dots,z_{p}) is an m×p𝑚𝑝m\times p design matrix, and β𝛽\beta is a p×1𝑝1p\times 1 vector of unknown regression coefficients. In the error term, η𝜂\eta is an unknown scalar and ϵ=(ϵ1,,ϵm)Titalic-ϵsuperscriptsubscriptitalic-ϵ1subscriptitalic-ϵ𝑚T\epsilon=(\epsilon_{1},\dots,\epsilon_{m})^{\mathrm{\scriptscriptstyle T}} has a spherically symmetric distribution,

ϵf~(ϵ2)similar-toitalic-ϵ~𝑓superscriptnormitalic-ϵ2\epsilon\sim\tilde{f}(\|\epsilon\|^{2}) (5.2)

where f~()~𝑓\tilde{f}(\cdot) is the probability density, E[ϵ]=0m𝐸delimited-[]italic-ϵsubscript0𝑚E[\epsilon]=0_{m}, and Var[ϵ]=ImVardelimited-[]italic-ϵsubscript𝐼𝑚\mbox{Var}[\epsilon]=I_{m}. Hence the density of y𝑦y is

yηm/2f~(ηyα1mZβ2),similar-to𝑦superscript𝜂𝑚2~𝑓𝜂superscriptnorm𝑦𝛼subscript1𝑚𝑍𝛽2y\sim\eta^{m/2}\tilde{f}(\eta\|y-\alpha 1_{m}-Z\beta\|^{2}), (5.3)

where f~~𝑓\tilde{f} satisfies

mf~(v2)dv=1subscriptsuperscript𝑚~𝑓superscriptnorm𝑣2differential-d𝑣1\int_{\mathbb{R}^{m}}\tilde{f}(\|v\|^{2})\mathrm{d}v=1

for v=(v1,,vm)Tm𝑣superscriptsubscript𝑣1subscript𝑣𝑚Tsuperscript𝑚v=(v_{1},\dots,v_{m})^{\mathrm{\scriptscriptstyle T}}\in\mathbb{R}^{m}. We assume that the columns of Z𝑍Z have been centered so that ziT1m=0superscriptsubscript𝑧𝑖Tsubscript1𝑚0z_{i}^{\mathrm{\scriptscriptstyle T}}1_{m}=0 for 1ip1𝑖𝑝1\leq i\leq p. We also assume that m>p+1𝑚𝑝1m>p+1 and {z1,,zp}subscript𝑧1subscript𝑧𝑝\{z_{1},\dots,z_{p}\} are linearly independent, which implies that

rankZ=p.rank𝑍𝑝\mbox{rank}\ Z=p.

Let Q𝑄Q be an m×m𝑚𝑚m\times m orthogonal matrix of the form

Q=(1m/m,Z(ZTZ)1/2,W)𝑄subscript1𝑚𝑚𝑍superscriptsuperscript𝑍T𝑍12𝑊Q=(1_{m}/\sqrt{m},Z(Z^{\mathrm{\scriptscriptstyle T}}Z)^{-1/2},W)

where W𝑊W is m×(mp1)𝑚𝑚𝑝1m\times(m-p-1) matrix which satisfies WT1m=0superscript𝑊Tsubscript1𝑚0W^{\mathrm{\scriptscriptstyle T}}1_{m}=0, WTZ=0superscript𝑊T𝑍0W^{\mathrm{\scriptscriptstyle T}}Z=0 and WTW=Imp1superscript𝑊T𝑊subscript𝐼𝑚𝑝1W^{\mathrm{\scriptscriptstyle T}}W=I_{m-p-1}. Also let x=(ZTZ)1/2ZTy=(ZTZ)1/2β^LSEp𝑥superscriptsuperscript𝑍T𝑍12superscript𝑍T𝑦superscriptsuperscript𝑍T𝑍12subscript^𝛽LSEsuperscript𝑝x=(Z^{\mathrm{\scriptscriptstyle T}}Z)^{-1/2}Z^{\mathrm{\scriptscriptstyle T}}y=(Z^{\mathrm{\scriptscriptstyle T}}Z)^{1/2}\hat{\beta}_{\mathrm{LSE}}\in\mathbb{R}^{p} where β^LSE=(ZTZ)1ZTysubscript^𝛽LSEsuperscriptsuperscript𝑍T𝑍1superscript𝑍T𝑦\hat{\beta}_{\mathrm{LSE}}=(Z^{\mathrm{\scriptscriptstyle T}}Z)^{-1}Z^{\mathrm{\scriptscriptstyle T}}y.

Let

QTy=(my¯,xT,uT)Tsuperscript𝑄T𝑦superscript𝑚¯𝑦superscript𝑥Tsuperscript𝑢TTQ^{\mathrm{\scriptscriptstyle T}}y=(\sqrt{m}\bar{y},x^{\mathrm{\scriptscriptstyle T}},u^{\mathrm{\scriptscriptstyle T}})^{\mathrm{\scriptscriptstyle T}}

where u=WTymp1𝑢superscript𝑊T𝑦superscript𝑚𝑝1u=W^{\mathrm{\scriptscriptstyle T}}y\in\mathbb{R}^{m-p-1}. Then (my¯,x,u)𝑚¯𝑦𝑥𝑢(\sqrt{m}\bar{y},x,u) are sufficient and the joint density of (my¯,x,u)𝑚¯𝑦𝑥𝑢(\sqrt{m}\bar{y},x,u) is

ηm/2f~(η{m(y¯α)2+xθ2+u2})superscript𝜂𝑚2~𝑓𝜂𝑚superscript¯𝑦𝛼2superscriptnorm𝑥𝜃2superscriptnorm𝑢2\eta^{m/2}\tilde{f}(\eta\{m(\bar{y}-\alpha)^{2}+\|x-\theta\|^{2}+\|u\|^{2}\})

where θ=(ZTZ)1/2β𝜃superscriptsuperscript𝑍T𝑍12𝛽\theta=(Z^{\mathrm{\scriptscriptstyle T}}Z)^{1/2}\beta. Further the marginal density of (x,u)𝑥𝑢(x,u) is

η(m1)/2f(η{xθ2+u2}),superscript𝜂𝑚12𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2\eta^{(m-1)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\}),

which we are considering in this paper, where m1=p+n𝑚1𝑝𝑛m-1=p+n and

f(t)=f~(v2+t)dv.𝑓𝑡superscriptsubscript~𝑓superscript𝑣2𝑡differential-d𝑣f(t)=\int_{-\infty}^{\infty}\tilde{f}(v^{2}+t)\mathrm{d}v.

Note that the loss function ηδθ2𝜂superscriptnorm𝛿𝜃2\eta\|\delta-\theta\|^{2} corresponds to so-called “predictive loss” ηZβ^Zβ2𝜂superscriptnorm𝑍^𝛽𝑍𝛽2\eta\|Z\hat{\beta}-Z\beta\|^{2} for estimation of the regression coefficient vector β𝛽\beta.

In the equivariant estimator δψsubscript𝛿𝜓\delta_{\psi} of θ𝜃\theta

{1ψ(x2/s)}x,1𝜓superscriptnorm𝑥2𝑠𝑥\left\{1-\psi(\|x\|^{2}/s)\right\}x,

x2/ssuperscriptnorm𝑥2𝑠\|x\|^{2}/s is R2/(1R2)superscript𝑅21superscript𝑅2R^{2}/(1-R^{2}) in the regression context where R2superscript𝑅2R^{2} is the coefficient of determination. It is natural to make use of R2superscript𝑅2R^{2} for shrinkage since small R2superscript𝑅2R^{2} corresponds to less reliability of the least squares estimator of β𝛽\beta. We note that the corresponding “simple Bayes estimator” for regression coefficient β𝛽\beta is rewritten as

(1a(a+1)(b+1)+R2/(1R2))β^LSE1𝑎𝑎1𝑏1superscript𝑅21superscript𝑅2subscript^𝛽LSE\left(1-\frac{a}{(a+1)(b+1)+R^{2}/(1-R^{2})}\right)\hat{\beta}_{\mathrm{LSE}}

and has a shrinkage factor which is increasing in R2superscript𝑅2R^{2}.

In the equivariant estimator δξ={1ξ(x/s)}x𝒟ξsubscript𝛿𝜉1𝜉𝑥𝑠𝑥subscript𝒟𝜉\delta_{\xi}=\left\{1-\xi(x/\sqrt{s})\right\}x\in\mathcal{D}_{\xi},

xs=(ZTZ)1/2β^LSEmp1σ^=(ZTZ)1/2mp1β^LSEσ^𝑥𝑠superscriptsuperscript𝑍T𝑍12subscript^𝛽LSE𝑚𝑝1^𝜎superscriptsuperscript𝑍T𝑍12𝑚𝑝1subscript^𝛽LSE^𝜎\frac{x}{\sqrt{s}}=\frac{(Z^{\mathrm{\scriptscriptstyle T}}Z)^{1/2}\hat{\beta}_{\mathrm{LSE}}}{\sqrt{m-p-1}\hat{\sigma}}=\frac{(Z^{\mathrm{\scriptscriptstyle T}}Z)^{1/2}}{\sqrt{m-p-1}}\frac{\hat{\beta}_{\mathrm{LSE}}}{\hat{\sigma}} (5.4)

where σ^=s/(mp1)^𝜎𝑠𝑚𝑝1\hat{\sigma}=\sqrt{s/(m-p-1)} and β^LSE/σ^subscript^𝛽LSE^𝜎\hat{\beta}_{\mathrm{LSE}}/\hat{\sigma} is a vector of the t𝑡t-values.

Hence the restriction to 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi} or 𝒟ξsubscript𝒟𝜉\mathcal{D}_{\xi} is quite natural in regression context. The minimaxity and admissibility results of Sections 3 and 4 provide some guidance as to reasonable shrinkage estimators in the regression context.

6 Concluding remarks

We have established admissibility of certain generalized Bayes estimators within the class of equivariant estimators, of the mean vector for a spherically symmetric distribution with unknown scale under invariant loss. In some cases, we establish simultaneous minimaxity and, equivariant admissibility for broader classes of sampling distributions. In the Gaussian case we establish admissibility within the equivariant estimators of a class of generalized Bayes minimax estimators of a particularly simple form. We have also investigated similar issues in the setting of a general linear regression model with intercept and spherically symmetric error distribution. In this setting, the shrinkage factor of equivariant estimators of the regression coefficients depends on the coefficient of determination.

Appendix A Proof of Theorem 2.1

Suppose the estimator δξ(X,S)𝒟ξsubscript𝛿𝜉𝑋𝑆subscript𝒟𝜉\delta_{\xi}(X,S)\in\mathcal{D}_{\xi} is strictly better than the estimator δψ𝒟ψsubscript𝛿𝜓subscript𝒟𝜓\delta_{\psi}\in\mathcal{D}_{\psi}, that is,

E[ηδξ(X,S)θ2]E[ηδψ(X,S)θ2]𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜉𝑋𝑆𝜃2𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜓𝑋𝑆𝜃2\displaystyle E\left[\eta\left\|\delta_{\xi}(X,S)-\theta\right\|^{2}\right]\leq E\left[\eta\left\|\delta_{\psi}(X,S)-\theta\right\|^{2}\right] (A.1)

for all η1/2θpsuperscript𝜂12𝜃superscript𝑝\eta^{1/2}\theta\in\mathbb{R}^{p} with strict inequality for some value. Because of the continuity of δξ(X,S)subscript𝛿𝜉𝑋𝑆\delta_{\xi}(X,S) and δψ(X,S)subscript𝛿𝜓𝑋𝑆\delta_{\psi}(X,S), strict inequality will hold for η1/2θpsuperscript𝜂12𝜃superscript𝑝\eta^{1/2}\theta\in\mathbb{R}^{p} in some nonempty open set Up𝑈superscript𝑝U\subset\mathbb{R}^{p}. The inequality (A.1) will remain true if δξ(X,S)subscript𝛿𝜉𝑋𝑆\delta_{\xi}(X,S) is replaced by Γδξ(Γ1X,S)Γsubscript𝛿𝜉superscriptΓ1𝑋𝑆\Gamma\delta_{\xi}(\Gamma^{-1}X,S) with ΓΓ\Gamma orthogonal, since

E[ηΓδξ(Γ1X,S)θ2]=E[ηδξ(Γ1X,S)Γ1θ2].𝐸delimited-[]𝜂superscriptnormΓsubscript𝛿𝜉superscriptΓ1𝑋𝑆𝜃2𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜉superscriptΓ1𝑋𝑆superscriptΓ1𝜃2\displaystyle E\left[\eta\left\|\Gamma\delta_{\xi}(\Gamma^{-1}X,S)-\theta\right\|^{2}\right]=E\left[\eta\left\|\delta_{\xi}(\Gamma^{-1}X,S)-\Gamma^{-1}\theta\right\|^{2}\right].

Thus, for fixed η1/2θUpsuperscript𝜂12𝜃𝑈superscript𝑝\eta^{1/2}\theta\in U\subset\mathbb{R}^{p}, the set of ΓΓ\Gamma for which

E[ηΓδξ(Γ1X,S)θ2]<E[ηδψ(X,S)θ2]𝐸delimited-[]𝜂superscriptnormΓsubscript𝛿𝜉superscriptΓ1𝑋𝑆𝜃2𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜓𝑋𝑆𝜃2\displaystyle E\left[\eta\left\|\Gamma\delta_{\xi}(\Gamma^{-1}X,S)-\theta\right\|^{2}\right]<E\left[\eta\left\|\delta_{\psi}(X,S)-\theta\right\|^{2}\right]

will be a nonempty open set. Let μ𝜇\mu be the invariant probability measure on 𝒪(p)𝒪𝑝\mathcal{O}(p) which assigns strictly positive measure to any nonempty open set (for the existence of such a measure, see Chapter 2 of Weil (1940)). Then the weighted estimator

δξ=𝒪(p)Γδξ(Γ1X,S)dμ(Γ)subscript𝛿𝜉subscript𝒪𝑝Γsubscript𝛿𝜉superscriptΓ1𝑋𝑆differential-d𝜇Γ\displaystyle\delta_{\xi\star}=\int_{\mathcal{O}(p)}\Gamma\delta_{\xi}(\Gamma^{-1}X,S)\mathrm{d}\mu(\Gamma)

is a member of the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}, and because of the convexity of the loss function in δ𝛿\delta, we have

E[ηδξ(X,S)θ2]𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜉𝑋𝑆𝜃2\displaystyle E\left[\eta\left\|\delta_{\xi\star}(X,S)-\theta\right\|^{2}\right] E[ηΓδξ(Γ1X,S)θ2]dμ(Γ)absent𝐸delimited-[]𝜂superscriptnormΓsubscript𝛿𝜉superscriptΓ1𝑋𝑆𝜃2differential-d𝜇Γ\displaystyle\leq\int E\left[\eta\left\|\Gamma\delta_{\xi}(\Gamma^{-1}X,S)-\theta\right\|^{2}\right]\mathrm{d}\mu(\Gamma)
E[ηδψ(X,S)θ2]absent𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜓𝑋𝑆𝜃2\displaystyle\leq E\left[\eta\left\|\delta_{\psi}(X,S)-\theta\right\|^{2}\right]

with strict inequality for η1/2θUsuperscript𝜂12𝜃𝑈\eta^{1/2}\theta\in U. This implies that δψ(X,S)subscript𝛿𝜓𝑋𝑆\delta_{\psi}(X,S) is not admissible among 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi} as assumed and hence completes the proof.

Appendix B Proof of Theorem 3.1

[Parts 1 and 2] The Bayes equivariant risk given by (3.4) is rewritten as

B(δψ,π)=pR~(μ2,δψ)π¯(μ2)dμ=pR~(ηθ2,δψ)ηp/2π¯(ηθ2)dθ=pR(θ,η,δψ)ηp/2π¯(ηθ2)dθ,𝐵subscript𝛿𝜓𝜋subscriptsuperscript𝑝~𝑅superscriptdelimited-∥∥𝜇2subscript𝛿𝜓¯𝜋superscriptdelimited-∥∥𝜇2differential-d𝜇subscriptsuperscript𝑝~𝑅𝜂superscriptdelimited-∥∥𝜃2subscript𝛿𝜓superscript𝜂𝑝2¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃subscriptsuperscript𝑝𝑅𝜃𝜂subscript𝛿𝜓superscript𝜂𝑝2¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃\begin{split}B(\delta_{\psi},\pi)&=\int_{\mathbb{R}^{p}}\tilde{R}(\|\mu\|^{2},\delta_{\psi})\bar{\pi}(\|\mu\|^{2})\mathrm{d}\mu\\ &=\int_{\mathbb{R}^{p}}\tilde{R}(\eta\|\theta\|^{2},\delta_{\psi})\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\\ &=\int_{\mathbb{R}^{p}}R(\theta,\eta,\delta_{\psi})\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta,\end{split} (B.1)

where the third equality follows from (3.2). Further B(δψ,π)𝐵subscript𝛿𝜓𝜋B(\delta_{\psi},\pi) given by (B.1) is expanded as

B(δψ,π)=pE[ηX2ψ2(X2/S)]ηp/2π¯(ηθ2)dθ2pE[ηX2ψ(X2/S)]ηp/2π¯(ηθ2)dθ+2pE[ηψ(X2/S)XTθ]ηp/2π¯(ηθ2)dθ+pE[ηXθ2]ηp/2π¯(ηθ2)dθ.𝐵subscript𝛿𝜓𝜋subscriptsuperscript𝑝𝐸delimited-[]𝜂superscriptdelimited-∥∥𝑋2superscript𝜓2superscriptdelimited-∥∥𝑋2𝑆superscript𝜂𝑝2¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃2subscriptsuperscript𝑝𝐸delimited-[]𝜂superscriptdelimited-∥∥𝑋2𝜓superscriptdelimited-∥∥𝑋2𝑆superscript𝜂𝑝2¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃2subscriptsuperscript𝑝𝐸delimited-[]𝜂𝜓superscriptdelimited-∥∥𝑋2𝑆superscript𝑋T𝜃superscript𝜂𝑝2¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃subscriptsuperscript𝑝𝐸delimited-[]𝜂superscriptdelimited-∥∥𝑋𝜃2superscript𝜂𝑝2¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃\begin{split}B(\delta_{\psi},\pi)&=\int_{\mathbb{R}^{p}}E\left[\eta\|X\|^{2}\psi^{2}(\|X\|^{2}/S)\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\\ &\qquad-2\int_{\mathbb{R}^{p}}E\left[\eta\|X\|^{2}\psi(\|X\|^{2}/S)\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\\ &\qquad+2\int_{\mathbb{R}^{p}}E\left[\eta\psi(\|X\|^{2}/S)X^{\mathrm{\scriptscriptstyle T}}\theta\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\\ &\qquad+\int_{\mathbb{R}^{p}}E\left[\eta\|X-\theta\|^{2}\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta.\end{split} (B.2)

Note that, by (1.5) and the propriety of the prior given by (3.7), the third term is equal to p𝑝p, that is,

pE[ηXθ2]ηp/2π¯(ηθ2)dθ=ppπ¯(μ2)dμ=p.subscriptsuperscript𝑝𝐸delimited-[]𝜂superscriptnorm𝑋𝜃2superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃subscriptsuperscript𝑝𝑝¯𝜋superscriptnorm𝜇2differential-d𝜇𝑝\int_{\mathbb{R}^{p}}E\left[\eta\|X-\theta\|^{2}\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta=\int_{\mathbb{R}^{p}}p\bar{\pi}(\|\mu\|^{2})\mathrm{d}\mu=p. (B.3)

The first and second terms of (B.2) with ψj(X2/S)superscript𝜓𝑗superscriptnorm𝑋2𝑆\psi^{j}(\|X\|^{2}/S) for j=2,1𝑗21j=2,1 respectively, are rewritten as

pE[ηX2ψj(X2/S)]ηp/2π¯(ηθ2)dθsubscriptsuperscript𝑝𝐸delimited-[]𝜂superscriptnorm𝑋2superscript𝜓𝑗superscriptnorm𝑋2𝑆superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃\displaystyle\int_{\mathbb{R}^{p}}E\left[\eta\|X\|^{2}\psi^{j}(\|X\|^{2}/S)\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta (B.4)
=cnηx2ψj(x2/s)η(2p+n)/2sn/21f(η{xθ2+s})absentsubscript𝑐𝑛triple-integral𝜂superscriptnorm𝑥2superscript𝜓𝑗superscriptnorm𝑥2𝑠superscript𝜂2𝑝𝑛2superscript𝑠𝑛21𝑓𝜂superscriptnorm𝑥𝜃2𝑠\displaystyle=c_{n}\iiint\eta\|x\|^{2}\psi^{j}(\|x\|^{2}/s)\eta^{(2p+n)/2}s^{n/2-1}f(\eta\{\|x-\theta\|^{2}+s\})
×π¯(ηθ2)dθdxdsabsent¯𝜋𝜂superscriptnorm𝜃2d𝜃d𝑥d𝑠\displaystyle\qquad\times\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}x\mathrm{d}s
=cnηsz2ψj(z2)η(2p+n)/2s(p+n)/21f(η{szθ2+s})absentsubscript𝑐𝑛triple-integral𝜂𝑠superscriptnorm𝑧2superscript𝜓𝑗superscriptnorm𝑧2superscript𝜂2𝑝𝑛2superscript𝑠𝑝𝑛21𝑓𝜂superscriptnorm𝑠𝑧𝜃2𝑠\displaystyle=c_{n}\iiint\eta s\|z\|^{2}\psi^{j}(\|z\|^{2})\eta^{(2p+n)/2}s^{(p+n)/2-1}f(\eta\{\|\sqrt{s}z-\theta\|^{2}+s\})
×π¯(ηθ2)dθdzds(z=x/s,J=sp/2)\displaystyle\qquad\times\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}z\mathrm{d}s\quad(z=x/\sqrt{s},\ J=s^{p/2})
=cnηsz2ψj(z2)η(2p+n)/2s(2p+n)/21f(sη{zθ2+1})absentsubscript𝑐𝑛triple-integral𝜂𝑠superscriptnorm𝑧2superscript𝜓𝑗superscriptnorm𝑧2superscript𝜂2𝑝𝑛2superscript𝑠2𝑝𝑛21𝑓𝑠𝜂superscriptnorm𝑧subscript𝜃21\displaystyle=c_{n}\iiint\eta s\|z\|^{2}\psi^{j}(\|z\|^{2})\eta^{(2p+n)/2}s^{(2p+n)/2-1}f(s\eta\{\|z-\theta_{*}\|^{2}+1\})
×π¯(ηsθ2)dθdzds(θ=θ/s,J=sp/2)\displaystyle\qquad\times\bar{\pi}(\eta s\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}z\mathrm{d}s\quad(\theta_{*}=\theta/\sqrt{s},\ J=s^{p/2})
=cnz2ψj(z2)η(2p+n)/2f(η{zθ2+1})absentsubscript𝑐𝑛triple-integralsuperscriptnorm𝑧2superscript𝜓𝑗superscriptnorm𝑧2superscriptsubscript𝜂2𝑝𝑛2𝑓subscript𝜂superscriptnorm𝑧subscript𝜃21\displaystyle=c_{n}\iiint\|z\|^{2}\psi^{j}(\|z\|^{2})\eta_{*}^{(2p+n)/2}f(\eta_{*}\{\|z-\theta_{*}\|^{2}+1\})
×π¯(ηθ2)dθdzdη(η=ηs,J=1/η)\displaystyle\qquad\times\bar{\pi}(\eta_{*}\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}z\mathrm{d}\eta_{*}\quad(\eta_{*}=\eta s,\ J=1/\eta)
=cnpz2ψj(z2)M1(z,π)dz,absentsubscript𝑐𝑛subscriptsuperscript𝑝superscriptnorm𝑧2superscript𝜓𝑗superscriptnorm𝑧2subscript𝑀1𝑧𝜋differential-d𝑧\displaystyle=c_{n}\int_{\mathbb{R}^{p}}\|z\|^{2}\psi^{j}(\|z\|^{2})M_{1}(z,\pi)\mathrm{d}z,

where cnsubscript𝑐𝑛c_{n} is given by (3.5), z=x/s𝑧𝑥𝑠z=x/\sqrt{s}, J𝐽J is the Jacobian, and

M1(z,π)=η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη.subscript𝑀1𝑧𝜋double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂M_{1}(z,\pi)=\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta. (B.5)

Similarly, the third term of (B.2) is rewritten as

pE[ηψ(X2/S)XTθ]ηp/2π¯(ηθ2)dθsubscriptsuperscript𝑝𝐸delimited-[]𝜂𝜓superscriptnorm𝑋2𝑆superscript𝑋T𝜃superscript𝜂𝑝2¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃\displaystyle\int_{\mathbb{R}^{p}}E\left[\eta\psi(\|X\|^{2}/S)X^{\mathrm{\scriptscriptstyle T}}\theta\right]\eta^{p/2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta (B.6)
=cnηψ2(x2/s)xTθη(2p+n)/2sn/21f(η{xθ2+s})absentsubscript𝑐𝑛triple-integral𝜂superscript𝜓2superscriptnorm𝑥2𝑠superscript𝑥T𝜃superscript𝜂2𝑝𝑛2superscript𝑠𝑛21𝑓𝜂superscriptnorm𝑥𝜃2𝑠\displaystyle=c_{n}\iiint\eta\psi^{2}(\|x\|^{2}/s)x^{\mathrm{\scriptscriptstyle T}}\theta\eta^{(2p+n)/2}s^{n/2-1}f(\eta\{\|x-\theta\|^{2}+s\})
×π¯(ηθ2)dθdxdsabsent¯𝜋𝜂superscriptnorm𝜃2d𝜃d𝑥d𝑠\displaystyle\qquad\times\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}x\mathrm{d}s
=cnηψ(z2)szTθη(2p+n)/2s(p+n)/21f(η{szθ2+s})absentsubscript𝑐𝑛triple-integral𝜂𝜓superscriptnorm𝑧2𝑠superscript𝑧T𝜃superscript𝜂2𝑝𝑛2superscript𝑠𝑝𝑛21𝑓𝜂superscriptnorm𝑠𝑧𝜃2𝑠\displaystyle=c_{n}\iiint\eta\psi(\|z\|^{2})\sqrt{s}z^{\mathrm{\scriptscriptstyle T}}\theta\eta^{(2p+n)/2}s^{(p+n)/2-1}f(\eta\{\|\sqrt{s}z-\theta\|^{2}+s\})
×π¯(ηθ2)dθdzds(z=x/s,J=sp/2)\displaystyle\qquad\times\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}z\mathrm{d}s\quad(z=x/\sqrt{s},\ J=s^{p/2})
=cnηsψ(z2)zTθη(2p+n)/2s(2p+n)/21f(sη{zθ2+1})absentsubscript𝑐𝑛triple-integral𝜂𝑠𝜓superscriptnorm𝑧2superscript𝑧Tsubscript𝜃superscript𝜂2𝑝𝑛2superscript𝑠2𝑝𝑛21𝑓𝑠𝜂superscriptnorm𝑧subscript𝜃21\displaystyle=c_{n}\iiint\eta s\psi(\|z\|^{2})z^{\mathrm{\scriptscriptstyle T}}\theta_{*}\eta^{(2p+n)/2}s^{(2p+n)/2-1}f(s\eta\{\|z-\theta_{*}\|^{2}+1\})
×π¯(ηsθ2)dθdzds(θ=θ/s,J=sp/2)\displaystyle\qquad\times\bar{\pi}(\eta s\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}z\mathrm{d}s\quad(\theta_{*}=\theta/\sqrt{s},\ J=s^{p/2})
=cnψ(z2)zTθη(2p+n)/2f(η{zθ2+1})absentsubscript𝑐𝑛triple-integral𝜓superscriptnorm𝑧2superscript𝑧Tsubscript𝜃superscriptsubscript𝜂2𝑝𝑛2𝑓subscript𝜂superscriptnorm𝑧subscript𝜃21\displaystyle=c_{n}\iiint\psi(\|z\|^{2})z^{\mathrm{\scriptscriptstyle T}}\theta_{*}\eta_{*}^{(2p+n)/2}f(\eta_{*}\{\|z-\theta_{*}\|^{2}+1\})
×π¯(ηθ2)dθdzdη(η=ηs,J=1/η)\displaystyle\qquad\times\bar{\pi}(\eta_{*}\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}z\mathrm{d}\eta_{*}\quad(\eta_{*}=\eta s,\ J=1/\eta)
=cnpψ(z2)zTM2(z,π)dz,absentsubscript𝑐𝑛subscriptsuperscript𝑝𝜓superscriptnorm𝑧2superscript𝑧Tsubscript𝑀2𝑧𝜋differential-d𝑧\displaystyle=c_{n}\int_{\mathbb{R}^{p}}\psi(\|z\|^{2})z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)\mathrm{d}z,

where

M2(z,π)=θη(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη.subscript𝑀2𝑧𝜋double-integral𝜃superscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂M_{2}(z,\pi)=\iint\theta\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta. (B.7)

Hence, by (B.3), (B.4) and (B.6), we have

B(δψ,π)=cnp{ψ2(z2)z2M1(z,π)2ψ(z2){z2M1(z,π)zTM2(z,π)}}dz+p.𝐵subscript𝛿𝜓𝜋subscript𝑐𝑛subscriptsuperscript𝑝superscript𝜓2superscriptdelimited-∥∥𝑧2superscriptdelimited-∥∥𝑧2subscript𝑀1𝑧𝜋2𝜓superscriptdelimited-∥∥𝑧2superscriptdelimited-∥∥𝑧2subscript𝑀1𝑧𝜋superscript𝑧Tsubscript𝑀2𝑧𝜋d𝑧𝑝\begin{split}B(\delta_{\psi},\pi)&=c_{n}\int_{\mathbb{R}^{p}}\left\{\psi^{2}(\|z\|^{2})\|z\|^{2}M_{1}(z,\pi)\right.\\ &\qquad\left.-2\psi(\|z\|^{2})\{\|z\|^{2}M_{1}(z,\pi)-z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)\}\right\}\mathrm{d}z+p.\end{split} (B.8)

Then the Bayes equivariant solution or minimizer of B(δψ,π)𝐵subscript𝛿𝜓𝜋B(\delta_{\psi},\pi) is

ψπ(z2)=argminψ(B(δψ,π))=1zTM2(z,π)z2M1(z,π)subscript𝜓𝜋superscriptnorm𝑧2subscriptargmin𝜓𝐵subscript𝛿𝜓𝜋1superscript𝑧Tsubscript𝑀2𝑧𝜋superscriptnorm𝑧2subscript𝑀1𝑧𝜋\psi_{\pi}(\|z\|^{2})=\operatorname*{arg\,min}_{\psi}\left(B(\delta_{\psi},\pi)\right)=1-\frac{z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)}{\|z\|^{2}M_{1}(z,\pi)} (B.9)

and hence the corresponding Bayes equivariant estimator is

δπ=ZTM2(Z,π)Z2M1(Z,π)X,subscript𝛿𝜋superscript𝑍Tsubscript𝑀2𝑍𝜋superscriptnorm𝑍2subscript𝑀1𝑍𝜋𝑋\displaystyle\delta_{\pi}=\frac{Z^{\mathrm{\scriptscriptstyle T}}M_{2}(Z,\pi)}{\|Z\|^{2}M_{1}(Z,\pi)}X, (B.10)

where Z=X/S𝑍𝑋𝑆Z=X/\sqrt{S}.

[Part 3] The generalized Bayes estimator of θ𝜃\theta with respect to the density on (θ,η)𝜃𝜂(\theta,\eta),

ηνηp/2g(ηθ2)superscript𝜂𝜈superscript𝜂𝑝2𝑔𝜂superscriptnorm𝜃2\eta^{\nu}\eta^{p/2}g(\eta\|\theta\|^{2})

is given by

δg,νsubscript𝛿𝑔𝜈\displaystyle\delta_{g,\nu} =E[ηθx,s]E[ηx,s]absent𝐸delimited-[]conditional𝜂𝜃𝑥𝑠𝐸delimited-[]conditional𝜂𝑥𝑠\displaystyle=\frac{E[\eta\theta\mid x,s]}{E[\eta\mid x,s]}
=ηθcnη(p+n)/2sn/21f(η{xθ2+s})ηνηp/2g(ηθ2)dθdηηcnη(p+n)/2sn/21f(η{xθ2+s})ηνηp/2g(ηθ2)dθdηabsentdouble-integral𝜂𝜃subscript𝑐𝑛superscript𝜂𝑝𝑛2superscript𝑠𝑛21𝑓𝜂superscriptnorm𝑥𝜃2𝑠superscript𝜂𝜈superscript𝜂𝑝2𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integral𝜂subscript𝑐𝑛superscript𝜂𝑝𝑛2superscript𝑠𝑛21𝑓𝜂superscriptnorm𝑥𝜃2𝑠superscript𝜂𝜈superscript𝜂𝑝2𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\frac{\iint\eta\theta c_{n}\eta^{(p+n)/2}s^{n/2-1}f(\eta\{\|x-\theta\|^{2}+s\})\eta^{\nu}\eta^{p/2}g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta c_{n}\eta^{(p+n)/2}s^{n/2-1}f(\eta\{\|x-\theta\|^{2}+s\})\eta^{\nu}\eta^{p/2}g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}
=θη(2p+n)/2+ν+1f(η{xθ2+s})g(ηθ2)dθdηη(2p+n)/2+ν+1f(η{xθ2+s})g(ηθ2)dθdη.absentdouble-integral𝜃superscript𝜂2𝑝𝑛2𝜈1𝑓𝜂superscriptnorm𝑥𝜃2𝑠𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝜈1𝑓𝜂superscriptnorm𝑥𝜃2𝑠𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\frac{\iint\theta\eta^{(2p+n)/2+\nu+1}f(\eta\{\|x-\theta\|^{2}+s\})g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2+\nu+1}f(\eta\{\|x-\theta\|^{2}+s\})g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}. (B.11)

By change of variables θ=θ/ssubscript𝜃𝜃𝑠\theta_{*}=\theta/\sqrt{s} and η=sηsubscript𝜂𝑠𝜂\eta_{*}=s\eta, we have

δg,ν=sθη(2p+n)/2+ν+1f(η{x/sθ2+1})g(ηθ2)dθdηη(2p+n)/2+ν+1f(η{x/sθ2+1})g(ηθ2)dθdη.subscript𝛿𝑔𝜈𝑠double-integralsubscript𝜃superscriptsubscript𝜂2𝑝𝑛2𝜈1𝑓subscript𝜂superscriptnorm𝑥𝑠subscript𝜃21𝑔subscript𝜂superscriptnormsubscript𝜃2differential-dsubscript𝜃differential-dsubscript𝜂double-integralsuperscriptsubscript𝜂2𝑝𝑛2𝜈1𝑓subscript𝜂superscriptnorm𝑥𝑠subscript𝜃21𝑔subscript𝜂superscriptnormsubscript𝜃2differential-dsubscript𝜃differential-dsubscript𝜂\displaystyle\delta_{g,\nu}=\sqrt{s}\frac{\iint\theta_{*}\eta_{*}^{(2p+n)/2+\nu+1}f(\eta_{*}\{\|x/\sqrt{s}-\theta_{*}\|^{2}+1\})g(\eta_{*}\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}\eta_{*}}{\iint\eta_{*}^{(2p+n)/2+\nu+1}f(\eta_{*}\{\|x/\sqrt{s}-\theta_{*}\|^{2}+1\})g(\eta_{*}\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}\eta_{*}}.

Comparing δg,νsubscript𝛿𝑔𝜈\delta_{g,\nu} with δπsubscript𝛿𝜋\delta_{\pi} given by (B.10), we see that δg,νsubscript𝛿𝑔𝜈\delta_{g,\nu} with ν=1𝜈1\nu=-1 is

δg,1=sM2(z,g)M1(z,g)=szzTM2(z,g)z2M1(z,g)=zTM2(z,g)z2M1(z,g)x.subscript𝛿𝑔1𝑠subscript𝑀2𝑧𝑔subscript𝑀1𝑧𝑔𝑠𝑧superscript𝑧Tsubscript𝑀2𝑧𝑔superscriptnorm𝑧2subscript𝑀1𝑧𝑔superscript𝑧Tsubscript𝑀2𝑧𝑔superscriptnorm𝑧2subscript𝑀1𝑧𝑔𝑥\delta_{g,-1}=\sqrt{s}\frac{M_{2}(z,g)}{M_{1}(z,g)}=\sqrt{s}\frac{zz^{\mathrm{\scriptscriptstyle T}}M_{2}(z,g)}{\|z\|^{2}M_{1}(z,g)}=\frac{z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,g)}{\|z\|^{2}M_{1}(z,g)}x.

The second equality follows since M2(z,g)subscript𝑀2𝑧𝑔M_{2}(z,g) is proportional to z𝑧z and the length of M2(z,g)subscript𝑀2𝑧𝑔M_{2}(z,g) is zTM2(z,g)/zsuperscript𝑧Tsubscript𝑀2𝑧𝑔norm𝑧z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,g)/\|z\|.

[Part 4] Since the quadratic loss function is strictly convex, the Bayes solution is unique, and hence admissibility within 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi} follows.

Appendix C Proof that δg,ν𝒟ψsubscript𝛿𝑔𝜈subscript𝒟𝜓\delta_{g,\nu}\in\mathcal{D}_{\psi}

As in (B.11), the generalized Bayes estimator of θ𝜃\theta with respect to ηνηp/2g(ηθ2)superscript𝜂𝜈superscript𝜂𝑝2𝑔𝜂superscriptnorm𝜃2\eta^{\nu}\eta^{p/2}g(\eta\|\theta\|^{2}) is given by

δg,ν(x,s)=θη(2p+n)/2+ν+1f(η{xθ2+s})g(ηθ2)dθdηη(2p+n)/2+ν+1f(η{xθ2+s})g(ηθ2)dθdη.subscript𝛿𝑔𝜈𝑥𝑠double-integral𝜃superscript𝜂2𝑝𝑛2𝜈1𝑓𝜂superscriptnorm𝑥𝜃2𝑠𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝜈1𝑓𝜂superscriptnorm𝑥𝜃2𝑠𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\delta_{g,\nu}(x,s)=\frac{\iint\theta\eta^{(2p+n)/2+\nu+1}f(\eta\{\|x-\theta\|^{2}+s\})g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2+\nu+1}f(\eta\{\|x-\theta\|^{2}+s\})g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}.

The estimator δg,ν(x,s)subscript𝛿𝑔𝜈𝑥𝑠\delta_{g,\nu}(x,s) with x=γΓx𝑥𝛾Γ𝑥x=\gamma\Gamma x and s=γ2s𝑠superscript𝛾2𝑠s=\gamma^{2}s is

δg,ν(γΓx,γ2s)=θη(2p+n)/2+ν+1f(η{γΓxθ2+γ2s})g(ηθ2)dθdηη(2p+n)/2+ν+1f(η{γΓxθ2+γ2s})g(ηθ2)dθdηsubscript𝛿𝑔𝜈𝛾Γ𝑥superscript𝛾2𝑠double-integral𝜃superscript𝜂2𝑝𝑛2𝜈1𝑓𝜂superscriptnorm𝛾Γ𝑥𝜃2superscript𝛾2𝑠𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝜈1𝑓𝜂superscriptnorm𝛾Γ𝑥𝜃2superscript𝛾2𝑠𝑔𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\delta_{g,\nu}(\gamma\Gamma x,\gamma^{2}s)=\frac{\iint\theta\eta^{(2p+n)/2+\nu+1}f(\eta\{\|\gamma\Gamma x-\theta\|^{2}+\gamma^{2}s\})g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2+\nu+1}f(\eta\{\|\gamma\Gamma x-\theta\|^{2}+\gamma^{2}s\})g(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}

and, by change of variables θ=γΓθ𝜃𝛾Γsubscript𝜃\theta=\gamma\Gamma\theta_{*} and η=γ2ηsubscript𝜂superscript𝛾2𝜂\eta_{*}=\gamma^{2}\eta, is rewritten as

δg,ν(γΓx,γ2s)subscript𝛿𝑔𝜈𝛾Γ𝑥superscript𝛾2𝑠\displaystyle\delta_{g,\nu}(\gamma\Gamma x,\gamma^{2}s) =γΓθη(p+n)/2+ν+1f(η{xθ2+s})g(ηθ2)dθdηη(p+n)/2+ν+1f(η{xθ2+s})g(ηθ2)dθdηabsent𝛾Γdouble-integralsubscript𝜃superscriptsubscript𝜂𝑝𝑛2𝜈1𝑓subscript𝜂superscriptnorm𝑥subscript𝜃2𝑠𝑔subscript𝜂superscriptnormsubscript𝜃2differential-dsubscript𝜃differential-dsubscript𝜂double-integralsuperscriptsubscript𝜂𝑝𝑛2𝜈1𝑓subscript𝜂superscriptnorm𝑥subscript𝜃2𝑠𝑔subscript𝜂superscriptnormsubscript𝜃2differential-dsubscript𝜃differential-dsubscript𝜂\displaystyle=\gamma\Gamma\frac{\iint\theta_{*}\eta_{*}^{(p+n)/2+\nu+1}f(\eta_{*}\{\|x-\theta_{*}\|^{2}+s\})g(\eta_{*}\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}\eta_{*}}{\iint\eta_{*}^{(p+n)/2+\nu+1}f(\eta_{*}\{\|x-\theta_{*}\|^{2}+s\})g(\eta_{*}\|\theta_{*}\|^{2})\mathrm{d}\theta_{*}\mathrm{d}\eta_{*}}
=γΓδg,ν(x,s).absent𝛾Γsubscript𝛿𝑔𝜈𝑥𝑠\displaystyle=\gamma\Gamma\delta_{g,\nu}(x,s).

Hence δg,ν𝒟ψsubscript𝛿𝑔𝜈subscript𝒟𝜓\delta_{g,\nu}\in\mathcal{D}_{\psi}.

Appendix D Proof of Theorem 4.1

Suppose that δπ𝒟ψsubscript𝛿𝜋subscript𝒟𝜓\delta_{\pi}\in\mathcal{D}_{\psi} is inadmissible among the class 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi} and hence δ𝒟ψsuperscript𝛿subscript𝒟𝜓\delta^{\prime}\in\mathcal{D}_{\psi} satisfies R~(λ,δ)R~(λ,δπ)~𝑅𝜆superscript𝛿~𝑅𝜆subscript𝛿𝜋\tilde{R}(\lambda,\delta^{\prime})\leq\tilde{R}(\lambda,\delta_{\pi}) for all λ𝜆\lambda with strict inequality for some λ𝜆\lambda. Let δ′′=(δπ+δ)/2superscript𝛿′′subscript𝛿𝜋superscript𝛿2\delta^{\prime\prime}=(\delta_{\pi}+\delta^{\prime})/2. Clearly δ′′superscript𝛿′′\delta^{\prime\prime} is also a member of 𝒟ψsubscript𝒟𝜓\mathcal{D}_{\psi}. Then, using Jensen’s inequality, we have

R~(λ,δ′′)~𝑅𝜆superscript𝛿′′\displaystyle\tilde{R}(\lambda,\delta^{\prime\prime}) =E[ηδ′′θ2]absent𝐸delimited-[]𝜂superscriptnormsuperscript𝛿′′𝜃2\displaystyle=E\left[\eta\|\delta^{\prime\prime}-\theta\|^{2}\right]
<(1/2)E[ηδθ2]+(1/2)E[ηδπθ2]absent12𝐸delimited-[]𝜂superscriptnormsuperscript𝛿𝜃212𝐸delimited-[]𝜂superscriptnormsubscript𝛿𝜋𝜃2\displaystyle<(1/2)E\left[\eta\|\delta^{\prime}-\theta\|^{2}\right]+(1/2)E\left[\eta\|\delta_{\pi}-\theta\|^{2}\right]
=12{R~(λ,δ)+R~(λ,δπ)}absent12~𝑅𝜆superscript𝛿~𝑅𝜆subscript𝛿𝜋\displaystyle=\frac{1}{2}\left\{\tilde{R}(\lambda,\delta^{\prime})+\tilde{R}(\lambda,\delta_{\pi})\right\}
R~(λ,δπ),absent~𝑅𝜆subscript𝛿𝜋\displaystyle\leq\tilde{R}(\lambda,\delta_{\pi}),

for any λ𝜆\lambda. Since R~(λ,δ′′)~𝑅𝜆superscript𝛿′′\tilde{R}(\lambda,\delta^{\prime\prime}) and R~(λ,δπ)~𝑅𝜆subscript𝛿𝜋\tilde{R}(\lambda,\delta_{\pi}) are both continuous functions of λ𝜆\lambda, there exists an ϵ>0italic-ϵ0\epsilon>0 such that R~(λ,δ′′)<R~(λ,δπ)ϵ~𝑅𝜆superscript𝛿′′~𝑅𝜆subscript𝛿𝜋italic-ϵ\tilde{R}(\lambda,\delta^{\prime\prime})<\tilde{R}(\lambda,\delta_{\pi})-\epsilon for 0λ10𝜆10\leq\lambda\leq 1. Then

diffB(δπ,δπi;πi)diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\displaystyle\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i}) =0{R~(λ,δπ)R~(λ,δπi)}πi(λ)dλabsentsuperscriptsubscript0~𝑅𝜆subscript𝛿𝜋~𝑅𝜆subscript𝛿𝜋𝑖subscript𝜋𝑖𝜆differential-d𝜆\displaystyle=\int_{0}^{\infty}\left\{\tilde{R}(\lambda,\delta_{\pi})-\tilde{R}(\lambda,\delta_{\pi i})\right\}\pi_{i}(\lambda)\mathrm{d}\lambda
0{R~(λ,δπ)R~(λ,δ′′)}πi(λ)dλabsentsuperscriptsubscript0~𝑅𝜆subscript𝛿𝜋~𝑅𝜆superscript𝛿′′subscript𝜋𝑖𝜆differential-d𝜆\displaystyle\geq\int_{0}^{\infty}\left\{\tilde{R}(\lambda,\delta_{\pi})-\tilde{R}(\lambda,\delta^{\prime\prime})\right\}\pi_{i}(\lambda)\mathrm{d}\lambda
01{R~(λ,δπ)R~(λ,δ′′)}π1(λ)dλabsentsuperscriptsubscript01~𝑅𝜆subscript𝛿𝜋~𝑅𝜆superscript𝛿′′subscript𝜋1𝜆differential-d𝜆\displaystyle\geq\int_{0}^{1}\left\{\tilde{R}(\lambda,\delta_{\pi})-\tilde{R}(\lambda,\delta^{\prime\prime})\right\}\pi_{1}(\lambda)\mathrm{d}\lambda
ϵγ>0,absentitalic-ϵ𝛾0\displaystyle\geq\epsilon\gamma>0,

which contradicts diffB(δπ,δπi;πi)0diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖0\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i})\to 0 as i𝑖i\to\infty.

Appendix E Preliminary results on π𝜋\pi, πisubscript𝜋𝑖\pi_{i} and f𝑓f

E.1 Preliminary results on π𝜋\pi

Lemma E.1.
  1. 1.

    Under Assumptions A.1A.3,

    supλ+λ|π(λ)|π(λ)subscriptsupremum𝜆subscript𝜆superscript𝜋𝜆𝜋𝜆\displaystyle\sup_{\lambda\in\mathbb{R}_{+}}\lambda\frac{|\pi^{\prime}(\lambda)|}{\pi(\lambda)}

    is bounded.

  2. 2.

    Under Assumption A.2, 01π(λ)λ1/2dλ<superscriptsubscript01𝜋𝜆superscript𝜆12differential-d𝜆\displaystyle\int_{0}^{1}\frac{\pi(\lambda)}{\lambda^{1/2}}\mathrm{d}\lambda<\infty.

  3. 3.

    Under Assumption A.2 with α>0𝛼0\alpha>0, 01π(λ)λdλ<superscriptsubscript01𝜋𝜆𝜆differential-d𝜆\displaystyle\int_{0}^{1}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda<\infty.

  4. 4.

    Under Assumption A.3A.3.1, 1π(λ)λdλ<superscriptsubscript1𝜋𝜆𝜆differential-d𝜆\displaystyle\int_{1}^{\infty}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda<\infty.

  5. 5.

    Under Assumption A.3, 1π(λ)λ2dλ<superscriptsubscript1𝜋𝜆superscript𝜆2differential-d𝜆\displaystyle\int_{1}^{\infty}\frac{\pi(\lambda)}{\lambda^{2}}\mathrm{d}\lambda<\infty.

  6. 6.

    If limλλπ(λ)/π(λ)<1subscript𝜆𝜆superscript𝜋𝜆𝜋𝜆1\lim_{\lambda\to\infty}\lambda\pi^{\prime}(\lambda)/\pi(\lambda)<-1, 1π(λ)dλ<superscriptsubscript1𝜋𝜆differential-d𝜆\displaystyle\int_{1}^{\infty}\pi(\lambda)\mathrm{d}\lambda<\infty.

  7. 7.

    Under Assumption A.3, there exist ϵ(0,1)italic-ϵ01\epsilon\in(0,1) and λ>exp(1)subscript𝜆1\lambda_{*}>\exp(1) such that π(λ)/{logλ}1ϵ𝜋𝜆superscript𝜆1italic-ϵ\pi(\lambda)/\{\log\lambda\}^{1-\epsilon} for λλ𝜆subscript𝜆\lambda\geq\lambda_{*} is bounded from above.

  8. 8.

    Under Assumption A.3A.3.2, 1π(λ)λκ2(λ)dλ<superscriptsubscript1𝜋𝜆𝜆superscript𝜅2𝜆differential-d𝜆\displaystyle\int_{1}^{\infty}\frac{\pi(\lambda)}{\lambda}\kappa^{2}(\lambda)\mathrm{d}\lambda<\infty.

Proof.

[Part 1] This follows from Assumptions A.1A.3 in a straightforward way.

[Part 2] We have

01π(λ)λ1/2dλsupλ[0,1]ν(λ)01λ1/2+α1dλ=supλ[0,1]ν(λ)11/2+α<.superscriptsubscript01𝜋𝜆superscript𝜆12differential-d𝜆subscriptsupremum𝜆01𝜈𝜆superscriptsubscript01superscript𝜆12𝛼1differential-d𝜆subscriptsupremum𝜆01𝜈𝜆112𝛼\int_{0}^{1}\frac{\pi(\lambda)}{\lambda^{1/2}}\mathrm{d}\lambda\leq\sup_{\lambda\in[0,1]}\nu(\lambda)\int_{0}^{1}\lambda^{1/2+\alpha-1}\mathrm{d}\lambda=\sup_{\lambda\in[0,1]}\nu(\lambda)\frac{1}{1/2+\alpha}<\infty. (E.1)

[Part 3] As in (E.1) of Part 2, we have

01π(λ)λdλsupλ[0,1]ν(λ)1α<.superscriptsubscript01𝜋𝜆𝜆differential-d𝜆subscriptsupremum𝜆01𝜈𝜆1𝛼\int_{0}^{1}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda\leq\sup_{\lambda\in[0,1]}\nu(\lambda)\frac{1}{\alpha}<\infty.

[Part 4] By Assumption A.3A.3.1, there exist ϵ>0italic-ϵ0\epsilon>0 and λ1>0subscript𝜆10\lambda_{1}>0 such that

λπ(λ)π(λ)ϵ,𝜆superscript𝜋𝜆𝜋𝜆italic-ϵ\displaystyle\lambda\frac{\pi^{\prime}(\lambda)}{\pi(\lambda)}\leq-\epsilon,

for all λλ1𝜆subscript𝜆1\lambda\geq\lambda_{1} and hence we have

λ1λπ(s)π(s)dsϵλ1λ1sdslogπ(λ)π(λ1)ϵlogλλ1superscriptsubscriptsubscript𝜆1𝜆superscript𝜋𝑠𝜋𝑠differential-d𝑠italic-ϵsuperscriptsubscriptsubscript𝜆1𝜆1𝑠differential-d𝑠𝜋𝜆𝜋subscript𝜆1italic-ϵ𝜆subscript𝜆1\int_{\lambda_{1}}^{\lambda}\frac{\pi^{\prime}(s)}{\pi(s)}\mathrm{d}s\leq-\epsilon\int_{\lambda_{1}}^{\lambda}\frac{1}{s}\mathrm{d}s\,\Leftrightarrow\,\log\frac{\pi(\lambda)}{\pi(\lambda_{1})}\leq-\epsilon\log\frac{\lambda}{\lambda_{1}} (E.2)

for λλ1𝜆subscript𝜆1\lambda\geq\lambda_{1}, which implies that

π(λ)π(λ1)λ1ϵλϵ for all λλ1.𝜋𝜆𝜋subscript𝜆1superscriptsubscript𝜆1italic-ϵsuperscript𝜆italic-ϵ for all 𝜆subscript𝜆1\pi(\lambda)\leq\frac{\pi(\lambda_{1})}{\lambda_{1}^{-\epsilon}}\lambda^{-\epsilon}\text{ for all }\lambda\geq\lambda_{1}. (E.3)

Hence we have

λ1π(λ)λdλπ(λ1)λ1ϵλ1dλλ1+ϵ=π(λ1)ϵ<.superscriptsubscriptsubscript𝜆1𝜋𝜆𝜆differential-d𝜆𝜋subscript𝜆1superscriptsubscript𝜆1italic-ϵsuperscriptsubscriptsubscript𝜆1d𝜆superscript𝜆1italic-ϵ𝜋subscript𝜆1italic-ϵ\int_{\lambda_{1}}^{\infty}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda\leq\frac{\pi(\lambda_{1})}{\lambda_{1}^{-\epsilon}}\int_{\lambda_{1}}^{\infty}\frac{\mathrm{d}\lambda}{\lambda^{1+\epsilon}}=\frac{\pi(\lambda_{1})}{\epsilon}<\infty. (E.4)

[Parts 5 and 6] The proof is omitted since it is similar to that of Part 4.

[Part 7] Under Assumptions A.3A.3.1, by (E.3), π(λ)0𝜋𝜆0\pi(\lambda)\to 0 as λ𝜆\lambda\to\infty. Under Assumption A.3(A.3.2)A.3.2.1, it is clear that π(λ)𝜋𝜆\pi(\lambda) is bounded. Under Assumption A.3(A.3.2)A.3.2.2, there exist ϵ(0,1)italic-ϵ01\epsilon\in(0,1) and λ2>exp(1)subscript𝜆21\lambda_{2}>\exp(1) such that

λπ(λ)π(λ)1ϵlogλ𝜆superscript𝜋𝜆𝜋𝜆1italic-ϵ𝜆\lambda\frac{\pi^{\prime}(\lambda)}{\pi(\lambda)}\leq\frac{1-\epsilon}{\log\lambda} (E.5)

for all λλ2𝜆subscript𝜆2\lambda\geq\lambda_{2}. As in (E.2) and (E.3), we have

λ2λπ(s)π(s)ds(1ϵ)λ2λdsslogssuperscriptsubscriptsubscript𝜆2𝜆superscript𝜋𝑠𝜋𝑠differential-d𝑠1italic-ϵsuperscriptsubscriptsubscript𝜆2𝜆d𝑠𝑠𝑠\displaystyle\int_{\lambda_{2}}^{\lambda}\frac{\pi^{\prime}(s)}{\pi(s)}\mathrm{d}s\leq(1-\epsilon)\int_{\lambda_{2}}^{\lambda}\frac{\mathrm{d}s}{s\log s}
logπ(λ)π(λ2)(1ϵ){loglogλloglogλ2}absent𝜋𝜆𝜋subscript𝜆21italic-ϵ𝜆subscript𝜆2\displaystyle\Leftrightarrow\log\frac{\pi(\lambda)}{\pi(\lambda_{2})}\leq(1-\epsilon)\left\{\log\log\lambda-\log\log\lambda_{2}\right\}

and hence

π(λ)π(λ2){logλ}1ϵ for all λλ2,𝜋𝜆𝜋subscript𝜆2superscript𝜆1italic-ϵ for all 𝜆subscript𝜆2\pi(\lambda)\leq\pi(\lambda_{2})\{\log\lambda\}^{1-\epsilon}\text{ for all }\lambda\geq\lambda_{2}, (E.6)

which completes the proof.

[Part 8] Under Assumption A.3(A.3.2)A.3.2.1, there exists λ3>0subscript𝜆30\lambda_{3}>0 such that |κ(λ)|𝜅𝜆|\kappa(\lambda)| for λλ3𝜆subscript𝜆3\lambda\geq\lambda_{3} is monotone decreasing. Then π(λ)𝜋𝜆\pi(\lambda) for λλ3𝜆subscript𝜆3\lambda\geq\lambda_{3} is expressed by

π(λ)=π(λ3)exp(λ3λ|κ(s)|sds)𝜋𝜆𝜋subscript𝜆3superscriptsubscriptsubscript𝜆3𝜆𝜅𝑠𝑠differential-d𝑠\displaystyle\pi(\lambda)=\pi(\lambda_{3})\exp\left(-\int_{\lambda_{3}}^{\lambda}\frac{|\kappa(s)|}{s}\mathrm{d}s\right)

and

λ3π(λ)λκ2(λ)dλsuperscriptsubscriptsubscript𝜆3𝜋𝜆𝜆superscript𝜅2𝜆differential-d𝜆\displaystyle\int_{\lambda_{3}}^{\infty}\frac{\pi(\lambda)}{\lambda}\kappa^{2}(\lambda)\mathrm{d}\lambda =π(λ3)λ3{κ(λ)}2λexp(λ3λ|κ(s)|sds)dλabsent𝜋subscript𝜆3superscriptsubscriptsubscript𝜆3superscript𝜅𝜆2𝜆superscriptsubscriptsubscript𝜆3𝜆𝜅𝑠𝑠differential-d𝑠differential-d𝜆\displaystyle=\pi(\lambda_{3})\int_{\lambda_{3}}^{\infty}\frac{\{\kappa(\lambda)\}^{2}}{\lambda}\exp\left(-\int_{\lambda_{3}}^{\lambda}\frac{|\kappa(s)|}{s}\mathrm{d}s\right)\mathrm{d}\lambda
π(λ3)|κ(λ3)|λ3|κ(λ)|λexp(λ3λ|κ(s)|sds)dλabsent𝜋subscript𝜆3𝜅subscript𝜆3superscriptsubscriptsubscript𝜆3𝜅𝜆𝜆superscriptsubscriptsubscript𝜆3𝜆𝜅𝑠𝑠differential-d𝑠differential-d𝜆\displaystyle\leq\pi(\lambda_{3})|\kappa(\lambda_{3})|\int_{\lambda_{3}}^{\infty}\frac{|\kappa(\lambda)|}{\lambda}\exp\left(-\int_{\lambda_{3}}^{\lambda}\frac{|\kappa(s)|}{s}\mathrm{d}s\right)\mathrm{d}\lambda
=π(λ3)|κ(λ3)|[exp(λ3λ|κ(s)|sds)]λ3absent𝜋subscript𝜆3𝜅subscript𝜆3superscriptsubscriptdelimited-[]superscriptsubscriptsubscript𝜆3𝜆𝜅𝑠𝑠differential-d𝑠subscript𝜆3\displaystyle=\pi(\lambda_{3})|\kappa(\lambda_{3})|\left[-\exp\left(-\int_{\lambda_{3}}^{\lambda}\frac{|\kappa(s)|}{s}\mathrm{d}s\right)\right]_{\lambda_{3}}^{\infty}
π(λ3)|κ(λ3)|<.absent𝜋subscript𝜆3𝜅subscript𝜆3\displaystyle\leq\pi(\lambda_{3})|\kappa(\lambda_{3})|<\infty.

Under Assumption A.3(A.3.2)A.3.2.2, by (E.6), we have

λ2π(λ)λκ2(λ)dλλ2π(λ2)λ{logλ}1+ϵdλ=π(λ2)ϵ{logλ2}ϵ<,superscriptsubscriptsubscript𝜆2𝜋𝜆𝜆superscript𝜅2𝜆differential-d𝜆superscriptsubscriptsubscript𝜆2𝜋subscript𝜆2𝜆superscript𝜆1italic-ϵdifferential-d𝜆𝜋subscript𝜆2italic-ϵsuperscriptsubscript𝜆2italic-ϵ\displaystyle\int_{\lambda_{2}}^{\infty}\frac{\pi(\lambda)}{\lambda}\kappa^{2}(\lambda)\mathrm{d}\lambda\leq\int_{\lambda_{2}}^{\infty}\frac{\pi(\lambda_{2})}{\lambda\{\log\lambda\}^{1+\epsilon}}\mathrm{d}\lambda=\frac{\pi(\lambda_{2})}{\epsilon\{\log\lambda_{2}\}^{\epsilon}}<\infty,

which completes the proof. ∎

Remark E.1.

By Parts 2 and 6 of Lemma E.1, if limλλπ(λ)/π(λ)<1subscript𝜆𝜆superscript𝜋𝜆𝜋𝜆1\lim_{\lambda\to\infty}\lambda\pi^{\prime}(\lambda)/\pi(\lambda)<-1, the prior π(λ)𝜋𝜆\pi(\lambda) with Assumption A.2 is proper and hence Part 4 of Theorem 3.1 can be applied. And this is why we assume limλλπ(λ)/π(λ)1subscript𝜆𝜆superscript𝜋𝜆𝜋𝜆1\lim_{\lambda\to\infty}\lambda\pi^{\prime}(\lambda)/\pi(\lambda)\geq-1 as the asymptotic behavior in Assumption A.3.

E.2 The sequence πisubscript𝜋𝑖\pi_{i}

The function hi(λ)subscript𝑖𝜆h_{i}(\lambda) in (4.5) satisfies the following.

Lemma E.2.
  1. 1.

    hi(λ)subscript𝑖𝜆h_{i}(\lambda) is increasing in i𝑖i for fixed λ𝜆\lambda, and decreasing in λ𝜆\lambda for fixed i𝑖i. Further limihi(λ)=1subscript𝑖subscript𝑖𝜆1\lim_{i\to\infty}h_{i}(\lambda)=1 for fixed λ0𝜆0\lambda\geq 0.

  2. 2.

    For fixed i𝑖i,

    limλ{(λ+e+i)log(λ+e+i)loglog(λ+e+i)}hi(λ)=i.subscript𝜆𝜆𝑒𝑖𝜆𝑒𝑖𝜆𝑒𝑖subscript𝑖𝜆𝑖\displaystyle\lim_{\lambda\to\infty}\{(\lambda+e+i)\log(\lambda+e+i)\log\log(\lambda+e+i)\}h_{i}(\lambda)=i.
  3. 3.

    For λ0𝜆0\lambda\geq 0,

    supi|hi(λ)|2(λ+e)log(λ+e)loglog(λ+e+1).subscriptsupremum𝑖subscriptsuperscript𝑖𝜆2𝜆𝑒𝜆𝑒𝜆𝑒1\displaystyle\sup_{i}|h^{\prime}_{i}(\lambda)|\leq\frac{2}{(\lambda+e)\log(\lambda+e)\log\log(\lambda+e+1)}.
  4. 4.

    h1(1)>1/8subscript1118h_{1}(1)>1/8.

  5. 5.

    supi,λ|hi(λ)|<5subscriptsupremum𝑖𝜆subscriptsuperscript𝑖𝜆5\sup_{i,\lambda}|h^{\prime}_{i}(\lambda)|<5.

  6. 6.

    Under Assumption A.2 on π𝜋\pi,

    01π1(λ)dλ>0.superscriptsubscript01subscript𝜋1𝜆differential-d𝜆0\displaystyle\int_{0}^{1}\pi_{1}(\lambda)\mathrm{d}\lambda>0.
  7. 7.

    Under Assumptions A.1, A.2 and A.3 on π𝜋\pi,

    0λπ(λ)supi{hi(λ)}2dλ<.superscriptsubscript0𝜆𝜋𝜆subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜆2d𝜆\displaystyle\int_{0}^{\infty}\lambda\pi(\lambda)\sup_{i}\{h^{\prime}_{i}(\lambda)\}^{2}\mathrm{d}\lambda<\infty.
  8. 8.

    Under Assumptions A.1, A.2 and A.3 on π𝜋\pi,

    0πi(λ)dλ<, for fixed i.superscriptsubscript0subscript𝜋𝑖𝜆differential-d𝜆 for fixed 𝑖\displaystyle\int_{0}^{\infty}\pi_{i}(\lambda)\mathrm{d}\lambda<\infty,\text{ for fixed }i.
Proof.

[Part 1] The part is straightforward given the form of hi(λ)subscript𝑖𝜆h_{i}(\lambda).

[Part 2] This follows from the expression,

hi(λ)subscript𝑖𝜆\displaystyle h_{i}(\lambda) =1loglog(λ+e+i)loglog(λ+e+i)log(λ+e)absent1𝜆𝑒𝑖𝜆𝑒𝑖𝜆𝑒\displaystyle=\frac{1}{\log\log(\lambda+e+i)}\log\frac{\log(\lambda+e+i)}{\log(\lambda+e)}
=log({λ+e+i}/{λ+e})log(λ+e+i)loglog(λ+e+i)ζ(log({λ+e+i}/{λ+e})log(λ+e+i))absent𝜆𝑒𝑖𝜆𝑒𝜆𝑒𝑖𝜆𝑒𝑖𝜁𝜆𝑒𝑖𝜆𝑒𝜆𝑒𝑖\displaystyle=\frac{\log(\{\lambda+e+i\}/\{\lambda+e\})}{\log(\lambda+e+i)\log\log(\lambda+e+i)}\zeta\left(\frac{\log(\{\lambda+e+i\}/\{\lambda+e\})}{\log(\lambda+e+i)}\right)
=i(λ+e+i)log(λ+e+i)loglog(λ+e+i)absent𝑖𝜆𝑒𝑖𝜆𝑒𝑖𝜆𝑒𝑖\displaystyle=\frac{i}{(\lambda+e+i)\log(\lambda+e+i)\log\log(\lambda+e+i)}
×ζ(iλ+e+i)ζ(log({λ+e+i}/{λ+e})log(λ+e+i)),absent𝜁𝑖𝜆𝑒𝑖𝜁𝜆𝑒𝑖𝜆𝑒𝜆𝑒𝑖\displaystyle\quad\times\zeta\left(\frac{i}{\lambda+e+i}\right)\zeta\left(\frac{\log(\{\lambda+e+i\}/\{\lambda+e\})}{\log(\lambda+e+i)}\right),

where ζ(x)=log(1x)/x𝜁𝑥1𝑥𝑥\zeta(x)=-\log(1-x)/x which satisfies limx0+ζ(x)=1subscript𝑥limit-from0𝜁𝑥1\lim_{x\to 0+}\zeta(x)=1.

[Part 3] The derivative is

hi(λ)subscriptsuperscript𝑖𝜆\displaystyle h^{\prime}_{i}(\lambda) =1(λ+e)log(λ+e)loglog(λ+e+i)absent1𝜆𝑒𝜆𝑒𝜆𝑒𝑖\displaystyle=-\frac{1}{(\lambda+e)\log(\lambda+e)\log\log(\lambda+e+i)}
+loglog(λ+e)(λ+e+i)log(λ+e+i){loglog(λ+e+i)}2.𝜆𝑒𝜆𝑒𝑖𝜆𝑒𝑖superscript𝜆𝑒𝑖2\displaystyle\quad+\frac{\log\log(\lambda+e)}{(\lambda+e+i)\log(\lambda+e+i)\{\log\log(\lambda+e+i)\}^{2}}.

Hence we have

|hi(λ)|subscriptsuperscript𝑖𝜆\displaystyle|h^{\prime}_{i}(\lambda)| |1(λ+e)log(λ+e)loglog(λ+e+i)|absent1𝜆𝑒𝜆𝑒𝜆𝑒𝑖\displaystyle\leq\left|\frac{1}{(\lambda+e)\log(\lambda+e)\log\log(\lambda+e+i)}\right|
+|loglog(λ+e)(λ+e+i)log(λ+e+i){loglog(λ+e+i)}2|𝜆𝑒𝜆𝑒𝑖𝜆𝑒𝑖superscript𝜆𝑒𝑖2\displaystyle\quad+\left|\frac{\log\log(\lambda+e)}{(\lambda+e+i)\log(\lambda+e+i)\{\log\log(\lambda+e+i)\}^{2}}\right|
2(λ+e)log(λ+e)loglog(λ+e+1)absent2𝜆𝑒𝜆𝑒𝜆𝑒1\displaystyle\leq\frac{2}{(\lambda+e)\log(\lambda+e)\log\log(\lambda+e+1)}

which does not depend on i𝑖i.

[Part 4] At λ=1𝜆1\lambda=1, h1(λ)subscript1𝜆h_{1}(\lambda) is

h1(1)=1loglog(1+e)loglog(2+e)=1e1+e1/(λlogλ)dλe2+e1/(λlogλ)dλ=1+e2+e1/(λlogλ)dλe2+e1/(λlogλ)dλ,subscript1111𝑒2𝑒1superscriptsubscript𝑒1𝑒1𝜆𝜆differential-d𝜆superscriptsubscript𝑒2𝑒1𝜆𝜆differential-d𝜆superscriptsubscript1𝑒2𝑒1𝜆𝜆differential-d𝜆superscriptsubscript𝑒2𝑒1𝜆𝜆differential-d𝜆\displaystyle h_{1}(1)=1-\frac{\log\log(1+e)}{\log\log(2+e)}=1-\frac{\int_{e}^{1+e}1/(\lambda\log\lambda)\mathrm{d}\lambda}{\int_{e}^{2+e}1/(\lambda\log\lambda)\mathrm{d}\lambda}=\frac{\int_{1+e}^{2+e}1/(\lambda\log\lambda)\mathrm{d}\lambda}{\int_{e}^{2+e}1/(\lambda\log\lambda)\mathrm{d}\lambda},

which is greater than

1/{(2+e)log(2+e)}2(1/e)>12e2+e1loge2>18.12𝑒2𝑒21𝑒12𝑒2𝑒1superscript𝑒218\frac{1/\{(2+e)\log(2+e)\}}{2(1/e)}>\frac{1}{2}\frac{e}{2+e}\frac{1}{\log e^{2}}>\frac{1}{8}.

[Part 5] The upper bound of supi|hi(λ)|subscriptsupremum𝑖subscriptsuperscript𝑖𝜆\sup_{i}|h^{\prime}_{i}(\lambda)|, derived in Part 3, is decreasing in λ𝜆\lambda and hence

supi|hi(λ)|supi|hi(λ)||λ=0=2eloglog(e+1)1loglog(e+1).subscriptsupremum𝑖subscriptsuperscript𝑖𝜆evaluated-atsubscriptsupremum𝑖subscriptsuperscript𝑖𝜆𝜆02𝑒𝑒11𝑒1\displaystyle\sup_{i}|h^{\prime}_{i}(\lambda)|\leq\sup_{i}|h^{\prime}_{i}(\lambda)|\big{|}_{\lambda=0}=\frac{2}{e\log\log(e+1)}\leq\frac{1}{\log\log(e+1)}.

Further we have

loglog(e+1)=ee+1dsslogs>log(e+1)log(e)log(e+1)=11log(e+1),𝑒1superscriptsubscript𝑒𝑒1d𝑠𝑠𝑠𝑒1𝑒𝑒111𝑒1\displaystyle\log\log(e+1)=\int_{e}^{e+1}\frac{\mathrm{d}s}{s\log s}>\frac{\log(e+1)-\log(e)}{\log(e+1)}=1-\frac{1}{\log(e+1)},
log(e+1)=log(e)+loge+1e=1log(11e+1)>1+1e+1,𝑒1𝑒𝑒1𝑒111𝑒111𝑒1\displaystyle\log(e+1)=\log(e)+\log\frac{e+1}{e}=1-\log\left(1-\frac{1}{e+1}\right)>1+\frac{1}{e+1},

and hence supλ,i|hi(λ)|e+2<5subscriptsupremum𝜆𝑖subscriptsuperscript𝑖𝜆𝑒25\sup_{\lambda,i}|h^{\prime}_{i}(\lambda)|\leq e+2<5.

[Part 6] By Parts 1 and 4, h12(λ)1/64subscriptsuperscript21𝜆164h^{2}_{1}(\lambda)\geq 1/64 for λ[0,1]𝜆01\lambda\in[0,1]. By Assumption A.2 on π𝜋\pi, there exists λ1(0,1)subscript𝜆101\lambda_{1}\in(0,1) such that π(λ)λα{ν(0)/2}𝜋𝜆superscript𝜆𝛼𝜈02\pi(\lambda)\geq\lambda^{\alpha}\{\nu(0)/2\} for λ[0,λ1]𝜆0subscript𝜆1\lambda\in[0,\lambda_{1}]. Then

01π(λ)h12(λ)dλν(0)21640λ1λαdλ=ν(0)λ1α+1128(α+1)>0.superscriptsubscript01𝜋𝜆subscriptsuperscript21𝜆differential-d𝜆𝜈02164superscriptsubscript0subscript𝜆1superscript𝜆𝛼differential-d𝜆𝜈0superscriptsubscript𝜆1𝛼1128𝛼10\displaystyle\int_{0}^{1}\pi(\lambda)h^{2}_{1}(\lambda)\mathrm{d}\lambda\geq\frac{\nu(0)}{2}\frac{1}{64}\int_{0}^{\lambda_{1}}\lambda^{\alpha}\mathrm{d}\lambda=\frac{\nu(0)\lambda_{1}^{\alpha+1}}{128(\alpha+1)}>0.

[Part 7] As in Part 7 of Lemma E.1, there exist ϵ(0,1)italic-ϵ01\epsilon\in(0,1) and λ2>exp(1)subscript𝜆21\lambda_{2}>\exp(1) such that

π(λ)π(λ2){logλ}1ϵ for all λλ2.𝜋𝜆𝜋subscript𝜆2superscript𝜆1italic-ϵ for all 𝜆subscript𝜆2\pi(\lambda)\leq\pi(\lambda_{2})\{\log\lambda\}^{1-\epsilon}\text{ for all }\lambda\geq\lambda_{2}. (E.7)

Then, by Part 5 and (E.7), we have

0λπ(λ)supi{hi(λ)}2dλsuperscriptsubscript0𝜆𝜋𝜆subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜆2d𝜆\displaystyle\int_{0}^{\infty}\lambda\pi(\lambda)\sup_{i}\{h^{\prime}_{i}(\lambda)\}^{2}\mathrm{d}\lambda
250λ2λπ(λ)dλ+λ2λπ(λ)supi{hi(λ)}2dλabsent25superscriptsubscript0subscript𝜆2𝜆𝜋𝜆differential-d𝜆superscriptsubscriptsubscript𝜆2𝜆𝜋𝜆subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜆2d𝜆\displaystyle\leq 25\int_{0}^{\lambda_{2}}\lambda\pi(\lambda)\mathrm{d}\lambda+\int_{\lambda_{2}}^{\infty}\lambda\pi(\lambda)\sup_{i}\{h^{\prime}_{i}(\lambda)\}^{2}\mathrm{d}\lambda
250λ2λπ(λ)dλ+π(λ2)λ24(λ+e)log(λ+e)dλ{(λ+e)log(λ+e)loglog(λ+e)}2absent25superscriptsubscript0subscript𝜆2𝜆𝜋𝜆differential-d𝜆𝜋subscript𝜆2superscriptsubscriptsubscript𝜆24𝜆𝑒𝜆𝑒d𝜆superscript𝜆𝑒𝜆𝑒𝜆𝑒2\displaystyle\leq 25\int_{0}^{\lambda_{2}}\lambda\pi(\lambda)\mathrm{d}\lambda+\pi(\lambda_{2})\int_{\lambda_{2}}^{\infty}\frac{4(\lambda+e)\log(\lambda+e)\mathrm{d}\lambda}{\left\{(\lambda+e)\log(\lambda+e)\log\log(\lambda+e)\right\}^{2}}
=250λ2λπ(λ)dλ+4π(λ2)loglog(λ2+e),absent25superscriptsubscript0subscript𝜆2𝜆𝜋𝜆differential-d𝜆4𝜋subscript𝜆2subscript𝜆2𝑒\displaystyle=25\int_{0}^{\lambda_{2}}\lambda\pi(\lambda)\mathrm{d}\lambda+\frac{4\pi(\lambda_{2})}{\log\log(\lambda_{2}+e)},

where 0λ2λπ(λ)dλsuperscriptsubscript0subscript𝜆2𝜆𝜋𝜆differential-d𝜆\int_{0}^{\lambda_{2}}\lambda\pi(\lambda)\mathrm{d}\lambda in the first term of the right-hand side is bounded by Part 2 of Lemma E.1.

[Part 8] The proof is omitted since it is similar to the proof of Part 7. ∎

E.3 Assumption on f𝑓f

Lemma E.3.

Let Assumptions F.1F.3 hold.

  1. 1.

    Also assume

    lim supttf(t)f(t)<p+n22jsubscriptlimit-supremum𝑡𝑡superscript𝑓𝑡𝑓𝑡𝑝𝑛22𝑗\limsup_{t\to\infty}\,t\frac{f^{\prime}(t)}{f(t)}<-\frac{p+n}{2}-2-j (E.8)

    for j0𝑗0j\geq 0 (hence j=0𝑗0j=0 for Assumption F.3F.3.1 and j=1𝑗1j=1 for Assumption F.3F.3.2).

    1. 1.A

      Then there exist ϵ(0,1)italic-ϵ01\epsilon\in(0,1) and t>1subscript𝑡1t_{*}>1 such that

      f(t)f(t)t(p+n)/22jϵt(p+n)/22jϵ,F(t)tf(t)(p+n)+2+2j+2ϵ,formulae-sequence𝑓𝑡𝑓subscript𝑡superscriptsubscript𝑡𝑝𝑛22𝑗italic-ϵsuperscript𝑡𝑝𝑛22𝑗italic-ϵ𝐹𝑡𝑡𝑓𝑡𝑝𝑛22𝑗2italic-ϵ\begin{split}f(t)&\leq\frac{f(t_{*})}{t_{*}^{-(p+n)/2-2-j-\epsilon}}t^{-(p+n)/2-2-j-\epsilon},\\ F(t)&\leq\frac{tf(t)}{(p+n)+2+2j+2\epsilon},\end{split} (E.9)

      for all tt𝑡subscript𝑡t\geq t_{*}, where

      F(t)=12tf(s)ds.𝐹𝑡12superscriptsubscript𝑡𝑓𝑠differential-d𝑠\displaystyle F(t)=\frac{1}{2}\int_{t}^{\infty}f(s)\mathrm{d}s.
    2. 1.B
      0t(p+n)/21+j{F(t)f(t)}2f(t)dt<.superscriptsubscript0superscript𝑡𝑝𝑛21𝑗superscript𝐹𝑡𝑓𝑡2𝑓𝑡differential-d𝑡\displaystyle\int_{0}^{\infty}t^{(p+n)/2-1+j}\left\{\frac{F(t)}{f(t)}\right\}^{2}f(t)\mathrm{d}t<\infty.
  2. 2.

    Assume Assumption F.3F.3.2. Also assume p3𝑝3p\geq 3. Let

    ~(t)~𝑡\displaystyle\tilde{\mathcal{F}}(t) =t1/2F(t)/f(t),absentsuperscript𝑡12𝐹𝑡𝑓𝑡\displaystyle=t^{1/2}F(t)/f(t),
    f(t)subscript𝑓𝑡\displaystyle f_{\star}(t) =0ηn/21~2(t+η)f(t+η)dη.absentsuperscriptsubscript0superscript𝜂𝑛21superscript~2𝑡𝜂𝑓𝑡𝜂differential-d𝜂\displaystyle=\int_{0}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(t+\eta)f(t+\eta)\mathrm{d}\eta.

    Then there exists 𝒬f>0subscript𝒬𝑓0\mathcal{Q}_{f}>0 such that

    p1y2f(yμ2)dy𝒬fmin(1,1/μ2).subscriptsuperscript𝑝1superscriptnorm𝑦2subscript𝑓superscriptnorm𝑦𝜇2differential-d𝑦subscript𝒬𝑓11superscriptnorm𝜇2\displaystyle\int_{\mathbb{R}^{p}}\frac{1}{\|y\|^{2}}f_{\star}(\|y-\mu\|^{2})\mathrm{d}y\leq\mathcal{Q}_{f}\min(1,1/\|\mu\|^{2}). (E.10)
Proof.

[Part 11.A] By (E.8), there exist t>1subscript𝑡1t_{*}>1 and ϵ(0,1)italic-ϵ01\epsilon\in(0,1) such that

tf(t)f(t)p+n22ϵj𝑡superscript𝑓𝑡𝑓𝑡𝑝𝑛22italic-ϵ𝑗t\frac{f^{\prime}(t)}{f(t)}\leq-\frac{p+n}{2}-2-\epsilon-j (E.11)

for all tt𝑡subscript𝑡t\geq t_{*}. Then, by (E.11), we have

ttf(s)f(s)ds(p+n22jϵ)ttdss for tt,formulae-sequencesuperscriptsubscriptsubscript𝑡𝑡superscript𝑓𝑠𝑓𝑠differential-d𝑠𝑝𝑛22𝑗italic-ϵsuperscriptsubscriptsubscript𝑡𝑡d𝑠𝑠 for 𝑡subscript𝑡\displaystyle\int_{t_{*}}^{t}\frac{f^{\prime}(s)}{f(s)}\mathrm{d}s\leq\left(-\frac{p+n}{2}-2-j-\epsilon\right)\int_{t_{*}}^{t}\frac{\mathrm{d}s}{s}\quad\text{ for }t\geq t_{*},
logf(t)f(t)(p+n22jϵ)logtt for tt,absentformulae-sequence𝑓𝑡𝑓subscript𝑡𝑝𝑛22𝑗italic-ϵ𝑡subscript𝑡 for 𝑡subscript𝑡\displaystyle\Leftrightarrow\log\frac{f(t)}{f(t_{*})}\leq\left(-\frac{p+n}{2}-2-j-\epsilon\right)\log\frac{t}{t_{*}}\quad\text{ for }t\geq t_{*},
f(t)f(t)t(p+n)/22jϵt(p+n)/22jϵ for tt.absentformulae-sequence𝑓𝑡𝑓subscript𝑡superscriptsubscript𝑡𝑝𝑛22𝑗italic-ϵsuperscript𝑡𝑝𝑛22𝑗italic-ϵ for 𝑡subscript𝑡\displaystyle\Leftrightarrow f(t)\leq\frac{f(t_{*})}{t_{*}^{-(p+n)/2-2-j-\epsilon}}t^{-(p+n)/2-2-j-\epsilon}\quad\text{ for }t\geq t_{*}. (E.12)

Further, by (E.11), we have

tf(t)(p+n2+2+j+ϵ)f(t),𝑡superscript𝑓𝑡𝑝𝑛22𝑗italic-ϵ𝑓𝑡\displaystyle tf^{\prime}(t)\leq-\left(\frac{p+n}{2}+2+j+\epsilon\right)f(t),

for all tt𝑡subscript𝑡t\geq t_{*}, and hence

tsf(s)ds(p+n2+2+j+ϵ)tf(s)ds.superscriptsubscript𝑡𝑠superscript𝑓𝑠differential-d𝑠𝑝𝑛22𝑗italic-ϵsuperscriptsubscript𝑡𝑓𝑠differential-d𝑠\int_{t}^{\infty}sf^{\prime}(s)\mathrm{d}s\leq-\left(\frac{p+n}{2}+2+j+\epsilon\right)\int_{t}^{\infty}f(s)\mathrm{d}s. (E.13)

By an integration by parts, the left-hand side is rewritten as

tsf(s)ds=[sf(s)]ttf(s)ds=tf(t)2F(t),superscriptsubscript𝑡𝑠superscript𝑓𝑠differential-d𝑠superscriptsubscriptdelimited-[]𝑠𝑓𝑠𝑡superscriptsubscript𝑡𝑓𝑠differential-d𝑠𝑡𝑓𝑡2𝐹𝑡\int_{t}^{\infty}sf^{\prime}(s)\mathrm{d}s=[sf(s)]_{t}^{\infty}-\int_{t}^{\infty}f(s)\mathrm{d}s=-tf(t)-2F(t),

where the second equality follows from [sf(s)]t=tf(t)superscriptsubscriptdelimited-[]𝑠𝑓𝑠𝑡𝑡𝑓𝑡[sf(s)]_{t}^{\infty}=-tf(t) by (E.12). Then the inequality (E.13) is equivalent to

tf(t)2F(t)2(p+n2+2+j+ϵ)F(t),F(t)f(t)t(p+n)+2+2j+2ϵ,formulae-sequence𝑡𝑓𝑡2𝐹𝑡2𝑝𝑛22𝑗italic-ϵ𝐹𝑡𝐹𝑡𝑓𝑡𝑡𝑝𝑛22𝑗2italic-ϵ\begin{split}-tf(t)-2F(t)&\leq-2\left(\frac{p+n}{2}+2+j+\epsilon\right)F(t),\\ \Leftrightarrow\quad\frac{F(t)}{f(t)}&\leq\frac{t}{(p+n)+2+2j+2\epsilon},\end{split} (E.14)

for all tt𝑡subscript𝑡t\geq t_{*}. Hence Part 11.A follows from (E.12) and (E.14).

[Part 11.B] By Assumption F.1, we have

01f(s)ds<.superscriptsubscript01𝑓𝑠differential-d𝑠\int_{0}^{1}f(s)\mathrm{d}s<\infty. (E.15)

Also the integrability given by (1.5),

1s(p+n)/21f(s)ds<,superscriptsubscript1superscript𝑠𝑝𝑛21𝑓𝑠differential-d𝑠\displaystyle\int_{1}^{\infty}s^{(p+n)/2-1}f(s)\mathrm{d}s<\infty,

implies

1f(s)ds<.superscriptsubscript1𝑓𝑠differential-d𝑠\int_{1}^{\infty}f(s)\mathrm{d}s<\infty. (E.16)

By (E.15) and (E.16), we have

F(0)=120f(s)ds<.𝐹012superscriptsubscript0𝑓𝑠differential-d𝑠F(0)=\frac{1}{2}\int_{0}^{\infty}f(s)\mathrm{d}s<\infty. (E.17)

Note 0<f(0)<0𝑓00<f(0)<\infty by Assumption F.1. Also by (E.14) and (E.17), it follows that there exists Cf>0subscript𝐶𝑓0C_{f}>0 such that

F(t)f(t)Cfmax(t,t),t0.formulae-sequence𝐹𝑡𝑓𝑡subscript𝐶𝑓𝑡subscript𝑡for-all𝑡0\frac{F(t)}{f(t)}\leq C_{f}\max(t,t_{*}),\quad\forall t\geq 0. (E.18)

By (E.18), for t[0,1]𝑡01t\in[0,1], we have

tj{F(t)f(t)}2f(t)Cf2t2maxt[0,1]f(t)superscript𝑡𝑗superscript𝐹𝑡𝑓𝑡2𝑓𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡2subscript𝑡01𝑓𝑡t^{j}\left\{\frac{F(t)}{f(t)}\right\}^{2}f(t)\leq C_{f}^{2}t_{*}^{2}\max_{t\in[0,1]}f(t) (E.19)

and hence

01t(p+n)/21+j{F(t)f(t)}2f(t)dt2Cf2t2p+nmaxt[0,1]f(t)<.superscriptsubscript01superscript𝑡𝑝𝑛21𝑗superscript𝐹𝑡𝑓𝑡2𝑓𝑡differential-d𝑡2superscriptsubscript𝐶𝑓2superscriptsubscript𝑡2𝑝𝑛subscript𝑡01𝑓𝑡\int_{0}^{1}t^{(p+n)/2-1+j}\left\{\frac{F(t)}{f(t)}\right\}^{2}f(t)\mathrm{d}t\leq\frac{2C_{f}^{2}t_{*}^{2}}{p+n}\max_{t\in[0,1]}f(t)<\infty. (E.20)

By (E.12) and (E.18), we have

tj{F(t)f(t)}2f(t)f(t)Cf2t(p+n)/22jϵt(p+n)/2ϵsuperscript𝑡𝑗superscript𝐹𝑡𝑓𝑡2𝑓𝑡𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛22𝑗italic-ϵsuperscript𝑡𝑝𝑛2italic-ϵt^{j}\left\{\frac{F(t)}{f(t)}\right\}^{2}f(t)\leq\frac{f(t_{*})C_{f}^{2}}{t_{*}^{-(p+n)/2-2-j-\epsilon}}t^{-(p+n)/2-\epsilon} (E.21)

for tt𝑡subscript𝑡t\geq t_{*} and hence

tt(p+n)/21+j{F(t)f(t)}2f(t)dtf(t)Cf2t(p+n)/22jϵtt1ϵdt=f(t)Cf2ϵt(p+n)/22j<.superscriptsubscriptsubscript𝑡superscript𝑡𝑝𝑛21𝑗superscript𝐹𝑡𝑓𝑡2𝑓𝑡differential-d𝑡𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛22𝑗italic-ϵsuperscriptsubscriptsubscript𝑡superscript𝑡1italic-ϵdifferential-d𝑡𝑓subscript𝑡superscriptsubscript𝐶𝑓2italic-ϵsuperscriptsubscript𝑡𝑝𝑛22𝑗\begin{split}\int_{t_{*}}^{\infty}t^{(p+n)/2-1+j}\left\{\frac{F(t)}{f(t)}\right\}^{2}f(t)\mathrm{d}t&\leq\frac{f(t_{*})C_{f}^{2}}{t_{*}^{-(p+n)/2-2-j-\epsilon}}\int_{t_{*}}^{\infty}t^{-1-\epsilon}\mathrm{d}t\\ &=\frac{f(t_{*})C_{f}^{2}}{\epsilon t_{*}^{-(p+n)/2-2-j}}<\infty.\end{split} (E.22)

Combining (E.20) and (E.22), completes the proof of Part 11.B.

[Part 2] Note, by Part 1 of this lemma with j=1𝑗1j=1,

0t(p+n)/21+1{F(t)f(t)}2f(t)dt=0t(p+n)/21~2(t)f(t)dt<.superscriptsubscript0superscript𝑡𝑝𝑛211superscript𝐹𝑡𝑓𝑡2𝑓𝑡differential-d𝑡superscriptsubscript0superscript𝑡𝑝𝑛21superscript~2𝑡𝑓𝑡differential-d𝑡\begin{split}\int_{0}^{\infty}t^{(p+n)/2-1+1}\left\{\frac{F(t)}{f(t)}\right\}^{2}f(t)\mathrm{d}t&=\int_{0}^{\infty}t^{(p+n)/2-1}\tilde{\mathcal{F}}^{2}(t)f(t)\mathrm{d}t\\ &<\infty.\end{split} (E.23)

To prove Part 2, it suffices to show that, for μ=0norm𝜇0\|\mu\|=0,

p1y2f(y2)dy<subscriptsuperscript𝑝1superscriptnorm𝑦2subscript𝑓superscriptnorm𝑦2differential-d𝑦\displaystyle\int_{\mathbb{R}^{p}}\frac{1}{\|y\|^{2}}f_{\star}(\|y\|^{2})\mathrm{d}y<\infty (E.24)

and also that there exist a>0𝑎0a>0 and b>0𝑏0b>0 such that

μ2p1y2f(yμ2)dy<bsuperscriptnorm𝜇2subscriptsuperscript𝑝1superscriptnorm𝑦2subscript𝑓superscriptnorm𝑦𝜇2differential-d𝑦𝑏\displaystyle\|\mu\|^{2}\int_{\mathbb{R}^{p}}\frac{1}{\|y\|^{2}}f_{\star}(\|y-\mu\|^{2})\mathrm{d}y<b (E.25)

for all μ2asuperscriptnorm𝜇2𝑎\|\mu\|^{2}\geq a.

[Bound in (E.24)] Note f(0)subscript𝑓0f_{\star}(0) is decomposed as

f(0)=0ηn/21~2(η)f(η)dη=01ηn/21~2(η)f(η)dη+1tηn/21~2(η)f(η)dη+tηn/21~2(η)f(η)dη,subscript𝑓0superscriptsubscript0superscript𝜂𝑛21superscript~2𝜂𝑓𝜂differential-d𝜂superscriptsubscript01superscript𝜂𝑛21superscript~2𝜂𝑓𝜂differential-d𝜂superscriptsubscript1subscript𝑡superscript𝜂𝑛21superscript~2𝜂𝑓𝜂differential-d𝜂superscriptsubscriptsubscript𝑡superscript𝜂𝑛21superscript~2𝜂𝑓𝜂differential-d𝜂\begin{split}f_{\star}(0)&=\int_{0}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\eta)f(\eta)\mathrm{d}\eta\\ &=\int_{0}^{1}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\eta)f(\eta)\mathrm{d}\eta+\int_{1}^{t_{*}}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\eta)f(\eta)\mathrm{d}\eta\\ &\qquad+\int_{t_{*}}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\eta)f(\eta)\mathrm{d}\eta,\end{split} (E.26)

where tsubscript𝑡t_{*} is from (E.11). The first and third terms both are integrable since, by (E.19),

01ηn/21~2(η)f(η)dηCf2t2maxη[0,1]f(η)01ηn/21dη=Cf2t22nmaxη[0,1]f(η),superscriptsubscript01superscript𝜂𝑛21superscript~2𝜂𝑓𝜂differential-d𝜂superscriptsubscript𝐶𝑓2superscriptsubscript𝑡2subscript𝜂01𝑓𝜂superscriptsubscript01superscript𝜂𝑛21differential-d𝜂superscriptsubscript𝐶𝑓2superscriptsubscript𝑡22𝑛subscript𝜂01𝑓𝜂\begin{split}\int_{0}^{1}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\eta)f(\eta)\mathrm{d}\eta&\leq C_{f}^{2}t_{*}^{2}\max_{\eta\in[0,1]}f(\eta)\int_{0}^{1}\eta^{n/2-1}\mathrm{d}\eta\\ &=C_{f}^{2}t_{*}^{2}\frac{2}{n}\max_{\eta\in[0,1]}f(\eta),\end{split} (E.27)

and by (E.21),

tηn/21~2(η)f(η)dηf(t)Cf2t(p+n)/23ϵtηn/21(p+n)/2ϵdη=f(t)Cf2t(p+n)/23ϵtp/2ϵp/2+ϵ.superscriptsubscriptsubscript𝑡superscript𝜂𝑛21superscript~2𝜂𝑓𝜂differential-d𝜂𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛23italic-ϵsuperscriptsubscriptsubscript𝑡superscript𝜂𝑛21𝑝𝑛2italic-ϵdifferential-d𝜂𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛23italic-ϵsuperscriptsubscript𝑡𝑝2italic-ϵ𝑝2italic-ϵ\begin{split}\int_{t_{*}}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\eta)f(\eta)\mathrm{d}\eta&\leq\frac{f(t_{*})C_{f}^{2}}{t_{*}^{-(p+n)/2-3-\epsilon}}\int_{t_{*}}^{\infty}\eta^{n/2-1-(p+n)/2-\epsilon}\mathrm{d}\eta\\ &=\frac{f(t_{*})C_{f}^{2}}{t_{*}^{-(p+n)/2-3-\epsilon}}\frac{t_{*}^{-p/2-\epsilon}}{p/2+\epsilon}.\end{split} (E.28)

By (E.26), (E.27) and (E.28), we have f(0)<subscript𝑓0f_{\star}(0)<\infty. Then, by continuity of fsubscript𝑓f_{\star}, it follows that

supt[0,1]f(t)<.subscriptsupremum𝑡01subscript𝑓𝑡\sup_{t\in[0,1]}f_{\star}(t)<\infty. (E.29)

Further the integrability of pf(y2)dysubscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦2differential-d𝑦\int_{\mathbb{R}^{p}}f_{\star}(\|y\|^{2})\mathrm{d}y follows since

pf(y2)dy=p0ηn/21~2(y2+η)f(y2+η)dηdy=1cnp+n~2(q2)f(q2)dq=cp+ncn0t(p+n)/21~2(t)f(t)dt<(by (E.23)).subscriptsuperscript𝑝subscript𝑓superscriptdelimited-∥∥𝑦2differential-d𝑦subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑛21superscript~2superscriptdelimited-∥∥𝑦2𝜂𝑓superscriptdelimited-∥∥𝑦2𝜂differential-d𝜂differential-d𝑦1subscript𝑐𝑛subscriptsuperscript𝑝𝑛superscript~2superscriptdelimited-∥∥𝑞2𝑓superscriptdelimited-∥∥𝑞2differential-d𝑞subscript𝑐𝑝𝑛subscript𝑐𝑛superscriptsubscript0superscript𝑡𝑝𝑛21superscript~2𝑡𝑓𝑡differential-d𝑡by (E.23)\begin{split}\int_{\mathbb{R}^{p}}f_{\star}(\|y\|^{2})\mathrm{d}y&=\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(\|y\|^{2}+\eta)f(\|y\|^{2}+\eta)\mathrm{d}\eta\mathrm{d}y\\ &=\frac{1}{c_{n}}\int_{\mathbb{R}^{p+n}}\tilde{\mathcal{F}}^{2}(\|q\|^{2})f(\|q\|^{2})\mathrm{d}q\\ &=\frac{c_{p+n}}{c_{n}}\int_{0}^{\infty}t^{(p+n)/2-1}\tilde{\mathcal{F}}^{2}(t)f(t)\mathrm{d}t\\ &<\infty\ (\text{by \eqref{eq:lem:f.2.1}}).\end{split} (E.30)

Then, by (E.29) and (E.30), we have

pf(y2)y2dysubscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦2superscriptnorm𝑦2differential-d𝑦\displaystyle\int_{\mathbb{R}^{p}}\frac{f_{\star}(\|y\|^{2})}{\|y\|^{2}}\mathrm{d}y supy1f(y2)y1dyy2+y1f(y2)dyabsentsubscriptsupremumnorm𝑦1subscript𝑓superscriptnorm𝑦2subscriptnorm𝑦1d𝑦superscriptnorm𝑦2subscriptnorm𝑦1subscript𝑓superscriptnorm𝑦2differential-d𝑦\displaystyle\leq\sup_{\|y\|\leq 1}f_{\star}(\|y\|^{2})\int_{\|y\|\leq 1}\frac{\mathrm{d}y}{\|y\|^{2}}+\int_{\|y\|\geq 1}f_{\star}(\|y\|^{2})\mathrm{d}y
2cpp2supy1f(y2)+pf(y2)dyabsent2subscript𝑐𝑝𝑝2subscriptsupremumnorm𝑦1subscript𝑓superscriptnorm𝑦2subscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦2differential-d𝑦\displaystyle\leq\frac{2c_{p}}{p-2}\sup_{\|y\|\leq 1}f_{\star}(\|y\|^{2})+\int_{\mathbb{R}^{p}}f_{\star}(\|y\|^{2})\mathrm{d}y
<.absent\displaystyle<\infty.

Hence the bound in (E.24) is established.

[Bound in (E.25)] Let μ2>2tsuperscriptnorm𝜇22subscript𝑡\|\mu\|^{2}>2t_{*} where tsubscript𝑡t_{*} is from (E.11). Under the decomposition of the integral region,

psuperscript𝑝\displaystyle\mathbb{R}^{p} ={y:yμ2μ2/2}absentconditional-set𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22\displaystyle=\left\{y:\|y-\mu\|^{2}\leq\|\mu\|^{2}/2\right\}
{y:yμ2μ2/2 and 0y2μ2}conditional-set𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22 and 0superscriptnorm𝑦2superscriptnorm𝜇2\displaystyle\qquad\cup\left\{y:\|y-\mu\|^{2}\geq\|\mu\|^{2}/2\text{ and }0\leq\|y\|^{2}\leq\|\mu\|^{2}\right\}
{y:yμ2μ2/2 and y2μ2}conditional-set𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22 and superscriptnorm𝑦2superscriptnorm𝜇2\displaystyle\qquad\cup\left\{y:\|y-\mu\|^{2}\geq\|\mu\|^{2}/2\text{ and }\|y\|^{2}\geq\|\mu\|^{2}\right\}
=123,absentsubscript1subscript2subscript3\displaystyle=\mathcal{R}_{1}\cup\mathcal{R}_{2}\cup\mathcal{R}_{3},

we have

μ2pf(yμ2)y2dysuperscriptnorm𝜇2subscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2differential-d𝑦\displaystyle\|\mu\|^{2}\int_{\mathbb{R}^{p}}\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y
=μ2(1+2+3)f(yμ2)y2dy.absentsuperscriptnorm𝜇2subscriptsubscript1subscriptsubscript2subscriptsubscript3subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2d𝑦\displaystyle=\|\mu\|^{2}\left(\int_{\mathcal{R}_{1}}+\int_{\mathcal{R}_{2}}+\int_{\mathcal{R}_{3}}\right)\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y.

For the region 1subscript1\mathcal{R}_{1}, yμ2μ2/2superscriptnorm𝑦𝜇2superscriptnorm𝜇22\|y-\mu\|^{2}\leq\|\mu\|^{2}/2 implies y2μ2/2superscriptnorm𝑦2superscriptnorm𝜇22\|y\|^{2}\geq\|\mu\|^{2}/2 and hence

μ21f(yμ2)y2dysuperscriptnorm𝜇2subscriptsubscript1subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2differential-d𝑦\displaystyle\|\mu\|^{2}\int_{\mathcal{R}_{1}}\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y 21f(yμ2)dyabsent2subscriptsubscript1subscript𝑓superscriptnorm𝑦𝜇2differential-d𝑦\displaystyle\leq 2\int_{\mathcal{R}_{1}}f_{\star}(\|y-\mu\|^{2})\mathrm{d}y
2pf(yμ2)dy,absent2subscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦𝜇2differential-d𝑦\displaystyle\leq 2\int_{\mathbb{R}^{p}}f_{\star}(\|y-\mu\|^{2})\mathrm{d}y,

which is bounded by (E.30). Similarly, for 1subscript1\mathcal{R}_{1}, since y2μ2superscriptnorm𝑦2superscriptnorm𝜇2\|y\|^{2}\geq\|\mu\|^{2}, we have

μ23f(yμ2)y2dypf(yμ2)dy<.superscriptnorm𝜇2subscriptsubscript3subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2differential-d𝑦subscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦𝜇2differential-d𝑦\displaystyle\|\mu\|^{2}\int_{\mathcal{R}_{3}}\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y\leq\int_{\mathbb{R}^{p}}f_{\star}(\|y-\mu\|^{2})\mathrm{d}y<\infty.

For the region

2={y:yμ2μ2/2 and 0y2μ2},subscript2conditional-set𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22 and 0superscriptnorm𝑦2superscriptnorm𝜇2\displaystyle\mathcal{R}_{2}=\left\{y:\|y-\mu\|^{2}\geq\|\mu\|^{2}/2\text{ and }0\leq\|y\|^{2}\leq\|\mu\|^{2}\right\},

we have

2{y:yμ2μ2/2},2{y:0y2μ2}.formulae-sequencesubscript2conditional-set𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22subscript2conditional-set𝑦0superscriptnorm𝑦2superscriptnorm𝜇2\displaystyle\mathcal{R}_{2}\subset\left\{y:\|y-\mu\|^{2}\geq\|\mu\|^{2}/2\right\},\quad\mathcal{R}_{2}\subset\left\{y:0\leq\|y\|^{2}\leq\|\mu\|^{2}\right\}.

Hence

μ22f(yμ2)y2dysupy:yμ2μ2/2f(yμ2)y:0y2μ2μ2y2dy,superscriptdelimited-∥∥𝜇2subscriptsubscript2subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2differential-d𝑦subscriptsupremum:𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22subscript𝑓superscriptdelimited-∥∥𝑦𝜇2subscript:𝑦0superscriptnorm𝑦2superscriptnorm𝜇2superscriptnorm𝜇2superscriptnorm𝑦2differential-d𝑦\begin{split}&\|\mu\|^{2}\int_{\mathcal{R}_{2}}\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y\\ &\leq\sup_{y:\|y-\mu\|^{2}\geq\|\mu\|^{2}/2}f_{\star}(\|y-\mu\|^{2})\int_{y:0\leq\|y\|^{2}\leq\|\mu\|^{2}}\frac{\|\mu\|^{2}}{\|y\|^{2}}\mathrm{d}y,\end{split} (E.31)

where

y:0y2μ2μ2y2dy=μ2cp0μ2rp/22dr=2cpp2μp.subscript:𝑦0superscriptnorm𝑦2superscriptnorm𝜇2superscriptnorm𝜇2superscriptnorm𝑦2differential-d𝑦superscriptnorm𝜇2subscript𝑐𝑝superscriptsubscript0superscriptnorm𝜇2superscript𝑟𝑝22differential-d𝑟2subscript𝑐𝑝𝑝2superscriptnorm𝜇𝑝\int_{y:0\leq\|y\|^{2}\leq\|\mu\|^{2}}\frac{\|\mu\|^{2}}{\|y\|^{2}}\mathrm{d}y=\|\mu\|^{2}c_{p}\int_{0}^{\|\mu\|^{2}}r^{p/2-2}\mathrm{d}r=\frac{2c_{p}}{p-2}\|\mu\|^{p}. (E.32)

Recall the assumption μ2>2tsuperscriptnorm𝜇22subscript𝑡\|\mu\|^{2}>2t_{*} and hence note

yμ2μ2/2>t.superscriptnorm𝑦𝜇2superscriptnorm𝜇22subscript𝑡\|y-\mu\|^{2}\geq\|\mu\|^{2}/2>t_{*}. (E.33)

By (E.21), the integrand of fsubscript𝑓f_{\star}, for tt𝑡subscript𝑡t\geq t_{*}, is bounded as

~2(t)f(t)f(t)Cf2t(p+n)/23ϵt(p+n)/2ϵsuperscript~2𝑡𝑓𝑡𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛23italic-ϵsuperscript𝑡𝑝𝑛2italic-ϵ\displaystyle\tilde{\mathcal{F}}^{2}(t)f(t)\leq\frac{f(t_{*})C_{f}^{2}}{t_{*}^{-(p+n)/2-3-\epsilon}}t^{-(p+n)/2-\epsilon}

and hence f(t)subscript𝑓𝑡f_{\star}(t) for tt𝑡subscript𝑡t\geq t_{*} is bounded as

f(t)=0ηn/21~2(t+η)f(t+η)dηf(t)Cf2t(p+n)/23ϵ0ηn/21(t+η)(p+n)/2ϵdη=C~ftp/2ϵ,subscript𝑓𝑡superscriptsubscript0superscript𝜂𝑛21superscript~2𝑡𝜂𝑓𝑡𝜂differential-d𝜂𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛23italic-ϵsuperscriptsubscript0superscript𝜂𝑛21superscript𝑡𝜂𝑝𝑛2italic-ϵdifferential-d𝜂subscript~𝐶𝑓superscript𝑡𝑝2italic-ϵ\begin{split}f_{\star}(t)&=\int_{0}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(t+\eta)f(t+\eta)\mathrm{d}\eta\\ &\leq\frac{f(t_{*})C_{f}^{2}}{t_{*}^{-(p+n)/2-3-\epsilon}}\int_{0}^{\infty}\eta^{n/2-1}(t+\eta)^{-(p+n)/2-\epsilon}\mathrm{d}\eta\\ &=\tilde{C}_{f}t^{-p/2-\epsilon},\end{split} (E.34)

where

C~f=f(t)Cf2t(p+n)/2+3+ϵB(p/2+ϵ,n/2).subscript~𝐶𝑓𝑓subscript𝑡superscriptsubscript𝐶𝑓2superscriptsubscript𝑡𝑝𝑛23italic-ϵ𝐵𝑝2italic-ϵ𝑛2\displaystyle\tilde{C}_{f}=f(t_{*})C_{f}^{2}t_{*}^{(p+n)/2+3+\epsilon}B(p/2+\epsilon,n/2).

Then, for any y{y:yμ2μ2/2}𝑦conditional-set𝑦superscriptnorm𝑦𝜇2superscriptnorm𝜇22y\in\{y:\|y-\mu\|^{2}\geq\|\mu\|^{2}/2\} with μ2>2tsuperscriptnorm𝜇22subscript𝑡\|\mu\|^{2}>2t_{*},

f(yμ2)C~f{yμ2}p/2ϵC~f2p/2+ϵμp2ϵ,subscript𝑓superscriptnorm𝑦𝜇2subscript~𝐶𝑓superscriptsuperscriptnorm𝑦𝜇2𝑝2italic-ϵsubscript~𝐶𝑓superscript2𝑝2italic-ϵsuperscriptnorm𝜇𝑝2italic-ϵf_{\star}(\|y-\mu\|^{2})\leq\tilde{C}_{f}\{\|y-\mu\|^{2}\}^{-p/2-\epsilon}\leq\frac{\tilde{C}_{f}}{2^{p/2+\epsilon}}\|\mu\|^{-p-2\epsilon}, (E.35)

where the first and second inequalities follow from (E.34) and (E.33), respectively. By (E.31), (E.32), and (E.35),

μ22f(yμ2)y2dy1μ2ϵ{C~f2p/2+ϵ2cpp2}superscriptnorm𝜇2subscriptsubscript2subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2differential-d𝑦1superscriptnorm𝜇2italic-ϵsubscript~𝐶𝑓superscript2𝑝2italic-ϵ2subscript𝑐𝑝𝑝2\displaystyle\|\mu\|^{2}\int_{\mathcal{R}_{2}}\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y\leq\frac{1}{\|\mu\|^{2\epsilon}}\left\{\tilde{C}_{f}2^{p/2+\epsilon}\frac{2c_{p}}{p-2}\right\}

which is bounded under the assumption μ2>2tsuperscriptnorm𝜇22subscript𝑡\|\mu\|^{2}>2t_{*}. ∎

Appendix F Preliminary results for completing Proof of Theorem 4.2

Note that the first three parts BL.1, BL.2 and BL.3 of Blyth’s (1951) conditions needed to prove Theorem 4.2 follow from Parts 1, 8 and 6 of Lemma E.2, respectively. In this appendix we provide an alternative expression diffB¯(z;δπ,δπi;πi)¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i}) in BL.4. The proof of Theorem 4.2’s two cases, I and II is completed in the two succeeding sections G and H respectively using this re-expression.

Recall as in (4.2) and (4.3),

diffB(δπ,δπi;πi)=cnpdiffB¯(z;δπ,δπi;πi)dz,diffB¯(z;δπ,δπi;πi)={ψπ(z2)ψπi(z2)}2z2M1(z,πi),formulae-sequencediff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖subscript𝑐𝑛subscriptsuperscript𝑝¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖differential-d𝑧¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖superscriptsubscript𝜓𝜋superscriptdelimited-∥∥𝑧2subscript𝜓𝜋𝑖superscriptdelimited-∥∥𝑧22superscriptdelimited-∥∥𝑧2subscript𝑀1𝑧subscript𝜋𝑖\begin{split}\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i})&=c_{n}\int_{\mathbb{R}^{p}}\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})\mathrm{d}z,\\ \overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})&=\{\psi_{\pi}(\|z\|^{2})-\psi_{\pi i}(\|z\|^{2})\}^{2}\|z\|^{2}M_{1}(z,\pi_{i}),\\ \end{split} (F.1)

with

ψπ(z)subscript𝜓𝜋𝑧\displaystyle\psi_{\pi}(z) =1zTM2(z,π)z2M1(z,π)=zTzM1(z,π)zTM2(z,π)z2M1(z,π),absent1superscript𝑧Tsubscript𝑀2𝑧𝜋superscriptnorm𝑧2subscript𝑀1𝑧𝜋superscript𝑧T𝑧subscript𝑀1𝑧𝜋superscript𝑧Tsubscript𝑀2𝑧𝜋superscriptnorm𝑧2subscript𝑀1𝑧𝜋\displaystyle=1-\frac{z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)}{\|z\|^{2}M_{1}(z,\pi)}=\frac{z^{\mathrm{\scriptscriptstyle T}}zM_{1}(z,\pi)-z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)}{\|z\|^{2}M_{1}(z,\pi)},
M1(z,π)subscript𝑀1𝑧𝜋\displaystyle M_{1}(z,\pi) =η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη,absentdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta,
M2(z,π)subscript𝑀2𝑧𝜋\displaystyle M_{2}(z,\pi) =θη(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη.absentdouble-integral𝜃superscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\iint\theta\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta.

The numerator of ψπ(z)subscript𝜓𝜋𝑧\psi_{\pi}(z) is rewritten as

zTzM1(z,π)zTM2(z,π)=zTη(zθ)η(2p+n)/21f(η{zθ2+1})π¯(ηθ2)dθdη=zTη(2p+n)/21θF(η{zθ2+1})π¯(ηθ2)dθdη=zTη(2p+n)/21F(η{zθ2+1})θπ¯(ηθ2)dθdη,superscript𝑧T𝑧subscript𝑀1𝑧𝜋superscript𝑧Tsubscript𝑀2𝑧𝜋superscript𝑧Tdouble-integral𝜂𝑧𝜃superscript𝜂2𝑝𝑛21𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂superscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21subscript𝜃𝐹𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂superscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptdelimited-∥∥𝑧𝜃21subscript𝜃¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂\begin{split}&z^{\mathrm{\scriptscriptstyle T}}zM_{1}(z,\pi)-z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)\\ &=z^{\mathrm{\scriptscriptstyle T}}\iint\eta(z-\theta)\eta^{(2p+n)/2-1}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta\\ &=z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}\nabla_{\theta}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta\\ &=-z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta,\end{split} (F.2)

where the last equality follows from an integration by parts. To justify this integration by parts, note that, for fixed θisubscript𝜃𝑖\theta_{i}, the i𝑖i-th component of θ𝜃\theta, we have

limθi±F(η{zθ2+1})π¯(ηθ2)=0subscriptsubscript𝜃𝑖plus-or-minus𝐹𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃20\lim_{\theta_{i}\to\pm\infty}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})=0

for any fixed η𝜂\eta, z𝑧z, θ1,,θi1,θi+1,,θpsubscript𝜃1subscript𝜃𝑖1subscript𝜃𝑖1subscript𝜃𝑝\theta_{1},\dots,\theta_{i-1},\theta_{i+1},\dots,\theta_{p}, since the asymptotic behavior of π¯¯𝜋\bar{\pi} and F𝐹F are given by

π¯(λ)=cp1λ1p/2π(λ)=o(λ1p/2logλ) and F(t)=o(t(p+n)/21),¯𝜋𝜆superscriptsubscript𝑐𝑝1superscript𝜆1𝑝2𝜋𝜆𝑜superscript𝜆1𝑝2𝜆 and 𝐹𝑡𝑜superscript𝑡𝑝𝑛21\displaystyle\bar{\pi}(\lambda)=c_{p}^{-1}\lambda^{1-p/2}\pi(\lambda)=o(\lambda^{1-p/2}\log\lambda)\text{ and }F(t)=o(t^{-(p+n)/2-1}),

as in Part 7 of Lemma E.1 and Part 11.A of Lemma E.3, respectively. Thus the last equality of (F.2) follows.

Therefore diffB¯(z;δπ,δπi;πi)¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i}) is re-expressed as

diffB¯(z;δπ,δπi;πi)=η(2p+n)/21F(η{zθ2+1})θπ¯(ηθ2)dθdηη(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdηη(2p+n)/21F(η{zθ2+1})θπ¯i(ηθ2)dθdηη(2p+n)/2f(η{zθ2+1})π¯i(ηθ2)dθdη2×η(2p+n)/2f(η{zθ2+1})π¯i(ηθ2)dθdη.\begin{split}&\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})\\ &=\left\|\frac{\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}\right.\\ &\qquad\left.-\frac{\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\nabla_{\theta}\bar{\pi}_{i}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}_{i}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}\right\|^{2}\\ &\qquad\qquad\times\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}_{i}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta.\end{split} (F.3)

The proof of Theorem 4.2, Cases I and II, will be completed in Sections G and H by showing diffB(δπ,δπi;πi)0diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖0\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i})\to 0 as i𝑖i\to\infty.

Appendix G Proof for Case I

By (F.3) and the decomposition

θπ¯i(ηθ2)subscript𝜃subscript¯𝜋𝑖𝜂superscriptnorm𝜃2\displaystyle\nabla_{\theta}\bar{\pi}_{i}(\eta\|\theta\|^{2}) =θ{π¯(ηθ2)hi2(ηθ2)}absentsubscript𝜃¯𝜋𝜂superscriptnorm𝜃2superscriptsubscript𝑖2𝜂superscriptnorm𝜃2\displaystyle=\nabla_{\theta}\left\{\bar{\pi}(\eta\|\theta\|^{2})h_{i}^{2}(\eta\|\theta\|^{2})\right\}
={θπ¯(ηθ2)}hi2(ηθ2)+π¯(ηθ2){θhi2(ηθ2)},absentsubscript𝜃¯𝜋𝜂superscriptnorm𝜃2superscriptsubscript𝑖2𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃2subscript𝜃superscriptsubscript𝑖2𝜂superscriptnorm𝜃2\displaystyle=\{\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\}h_{i}^{2}(\eta\|\theta\|^{2})+\bar{\pi}(\eta\|\theta\|^{2})\{\nabla_{\theta}h_{i}^{2}(\eta\|\theta\|^{2})\},

we have

diffB¯(z;δπ,δπi;πi)¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\displaystyle\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})
=cnη(2p+n)/21F()θπ¯()dθdηη(2p+n)/2f()π¯()dθdηη(2p+n)/21F()θπ¯()hi2()dθdηη(2p+n)/2f()π¯i()dθdηabsentconditionalsubscript𝑐𝑛double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋superscriptsubscript𝑖2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle=c_{n}\left\|\frac{\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}-\frac{\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right.
η(2p+n)/21F()π¯()θhi2()dθdηη(2p+n)/2f()π¯i()dθdη2η(2p+n)/2f()π¯i()dθdη,\displaystyle\qquad\left.-\frac{\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\nabla_{\theta}h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right\|^{2}\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta,

where, for notational convenience and to control the size of expressions,

=ηθ2,=η(zθ2+1).\displaystyle\bullet=\eta\|\theta\|^{2},\quad\circ=\eta(\|z-\theta\|^{2}+1).

Further, by the triangle inequality and the fact hi21superscriptsubscript𝑖21h_{i}^{2}\leq 1, we have

diffB¯(z;δπ,δπi;πi)2cn(Δ1i+Δ2i),¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖2subscript𝑐𝑛subscriptΔ1𝑖subscriptΔ2𝑖\displaystyle\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})\leq 2c_{n}(\Delta_{1i}+\Delta_{2i}),

where

Δ1isubscriptΔ1𝑖\displaystyle\Delta_{1i} =η(2p+n)/21F()π¯()θhi2()dθdη2η(2p+n)/2f()π¯i()dθdη,absentsuperscriptnormdouble-integralsuperscript𝜂2𝑝𝑛21𝐹¯𝜋subscript𝜃superscriptsubscript𝑖2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle=\frac{\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\nabla_{\theta}h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}, (G.1)
Δ2isubscriptΔ2𝑖\displaystyle\Delta_{2i} =η(2p+n)/21F()θπ¯()dθdη2η(2p+n)/2f()π¯()dθdηabsentsuperscriptnormdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle=\frac{\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}
+η(2p+n)/21F()θπ¯()hi2()dθdη2η(2p+n)/2f()π¯i()dθdη.superscriptnormdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋superscriptsubscript𝑖2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad+\frac{\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}. (G.2)

The proof of Case I will be completed by showing that each ΔjisubscriptΔ𝑗𝑖\Delta_{ji} for j=1,2𝑗12j=1,2 is bounded by an integrable function. The theorem then follows by the dominated convergence theorem since limidiffB(δπ,δπi;πi)=0subscript𝑖diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖0\lim_{i\to\infty}\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i})=0 since hi21superscriptsubscript𝑖21h_{i}^{2}\to 1 and δπiδπsubscript𝛿subscript𝜋𝑖subscript𝛿𝜋\delta_{\pi_{i}}\to\delta_{\pi} in the expression of (F.1).

G.1 Δ1isubscriptΔ1𝑖\Delta_{1i}

Note

θhi2(ηθ2)subscript𝜃superscriptsubscript𝑖2𝜂superscriptnorm𝜃2\displaystyle\nabla_{\theta}h_{i}^{2}(\eta\|\theta\|^{2}) =2hi(ηθ2)θhi(ηθ2),absent2subscript𝑖𝜂superscriptnorm𝜃2subscript𝜃subscript𝑖𝜂superscriptnorm𝜃2\displaystyle=2h_{i}(\eta\|\theta\|^{2})\nabla_{\theta}h_{i}(\eta\|\theta\|^{2}),
θhi(ηθ2)2superscriptnormsubscript𝜃subscript𝑖𝜂superscriptnorm𝜃22\displaystyle\|\nabla_{\theta}h_{i}(\eta\|\theta\|^{2})\|^{2} =4η2θ2{hi(ηθ2)}2.absent4superscript𝜂2superscriptnorm𝜃2superscriptsubscriptsuperscript𝑖𝜂superscriptnorm𝜃22\displaystyle=4\eta^{2}\|\theta\|^{2}\{h^{\prime}_{i}(\eta\|\theta\|^{2})\}^{2}.

Then, by Cauchy-Schwarz inequality,

Δ1isubscriptΔ1𝑖\displaystyle\Delta_{1i} =η(2p+n)/21F()π¯(){2hi()θhi()}dθdη2η(2p+n)/2f()π¯i()dθdηabsentsuperscriptnormdouble-integralsuperscript𝜂2𝑝𝑛21𝐹¯𝜋2subscript𝑖subscript𝜃subscript𝑖differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle=\frac{\|\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\{2h_{i}(\bullet)\nabla_{\theta}h_{i}(\bullet)\}\mathrm{d}\theta\mathrm{d}\eta\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}
4η(2p+n)/222()f()π¯()θhi()2dθdηabsent4double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓¯𝜋superscriptnormsubscript𝜃subscript𝑖2differential-d𝜃differential-d𝜂\displaystyle\leq 4\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\bar{\pi}(\bullet)\|\nabla_{\theta}h_{i}(\bullet)\|^{2}\mathrm{d}\theta\mathrm{d}\eta
=16η(2p+n)/222()f()π¯()η2θ2{hi()}2dθdη,absent16double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓¯𝜋superscript𝜂2superscriptnorm𝜃2superscriptsubscriptsuperscript𝑖2differential-d𝜃differential-d𝜂\displaystyle=16\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\bar{\pi}(\bullet)\eta^{2}\|\theta\|^{2}\{h^{\prime}_{i}(\bullet)\}^{2}\mathrm{d}\theta\mathrm{d}\eta,

where (t)=F(t)/f(t)𝑡𝐹𝑡𝑓𝑡\mathcal{F}(t)=F(t)/f(t). Then

psupiΔ1idz16subscriptsuperscript𝑝subscriptsupremum𝑖subscriptΔ1𝑖d𝑧16\displaystyle\frac{\int_{\mathbb{R}^{p}}\sup_{i}\Delta_{1i}\mathrm{d}z}{16} η(2p+n)/212(η{zθ2+1})f(η{zθ2+1})absenttriple-integralsuperscript𝜂2𝑝𝑛21superscript2𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21\displaystyle\leq\iiint\eta^{(2p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})
×ηθ2π¯(ηθ2)supi{hi(ηθ2)}2dθdηdzabsent𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃2subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜂superscriptnorm𝜃22d𝜃d𝜂d𝑧\displaystyle\qquad\times\eta\|\theta\|^{2}\bar{\pi}(\eta\|\theta\|^{2})\sup_{i}\{h^{\prime}_{i}(\eta\|\theta\|^{2})\}^{2}\mathrm{d}\theta\mathrm{d}\eta\mathrm{d}z
=η(p+n)/212(η{z2+1})f(η{z2+1})absenttriple-integralsuperscript𝜂𝑝𝑛21superscript2𝜂superscriptnorm𝑧21𝑓𝜂superscriptnorm𝑧21\displaystyle=\iiint\eta^{(p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z\|^{2}+1\})f(\eta\{\|z\|^{2}+1\})
×μ2π¯(μ2)supi{hi(μ2)}2dμdηdzabsentsuperscriptnorm𝜇2¯𝜋superscriptnorm𝜇2subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖superscriptnorm𝜇22d𝜇d𝜂d𝑧\displaystyle\qquad\times\|\mu\|^{2}\bar{\pi}(\|\mu\|^{2})\sup_{i}\{h^{\prime}_{i}(\|\mu\|^{2})\}^{2}\mathrm{d}\mu\mathrm{d}\eta\mathrm{d}z
A1cpwp/21η(p+n)/212(η{w+1})f(η{w+1})dwdηabsentsubscript𝐴1subscript𝑐𝑝double-integralsuperscript𝑤𝑝21superscript𝜂𝑝𝑛21superscript2𝜂𝑤1𝑓𝜂𝑤1differential-d𝑤differential-d𝜂\displaystyle\leq A_{1}c_{p}\iint w^{p/2-1}\eta^{(p+n)/2-1}\mathcal{F}^{2}(\eta\{w+1\})f(\eta\{w+1\})\mathrm{d}w\mathrm{d}\eta
=A1cp0wp/21dw(1+w)(p+n)/20t(p+n)/212(t)f(t)dtabsentsubscript𝐴1subscript𝑐𝑝superscriptsubscript0superscript𝑤𝑝21d𝑤superscript1𝑤𝑝𝑛2superscriptsubscript0superscript𝑡𝑝𝑛21superscript2𝑡𝑓𝑡differential-d𝑡\displaystyle=A_{1}c_{p}\int_{0}^{\infty}\frac{w^{p/2-1}\mathrm{d}w}{(1+w)^{(p+n)/2}}\int_{0}^{\infty}t^{(p+n)/2-1}\mathcal{F}^{2}(t)f(t)\mathrm{d}t
A1A2cpB(p/2,n/2),absentsubscript𝐴1subscript𝐴2subscript𝑐𝑝𝐵𝑝2𝑛2\displaystyle\leq A_{1}A_{2}c_{p}B(p/2,n/2),

where

A1subscript𝐴1\displaystyle A_{1} =pμ2π¯(μ2)supi{hi(μ2)}2dμabsentsubscriptsuperscript𝑝superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖superscriptnorm𝜇22d𝜇\displaystyle=\int_{\mathbb{R}^{p}}\|\mu\|^{2}\bar{\pi}(\|\mu\|^{2})\sup_{i}\{h^{\prime}_{i}(\|\mu\|^{2})\}^{2}\mathrm{d}\mu
=0λπ(λ)supi{hi(λ)}2dλabsentsuperscriptsubscript0𝜆𝜋𝜆subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜆2d𝜆\displaystyle=\int_{0}^{\infty}\lambda\pi(\lambda)\sup_{i}\{h^{\prime}_{i}(\lambda)\}^{2}\mathrm{d}\lambda (G.3)
A2subscript𝐴2\displaystyle A_{2} =0t(p+n)/212(t)f(t)dtabsentsuperscriptsubscript0superscript𝑡𝑝𝑛21superscript2𝑡𝑓𝑡differential-d𝑡\displaystyle=\int_{0}^{\infty}t^{(p+n)/2-1}\mathcal{F}^{2}(t)f(t)\mathrm{d}t (G.4)

are both bounded as shown in Part 7 of Lemma E.2 and in Part 1 of Lemma E.3, respectively.

G.2 Δ2isubscriptΔ2𝑖\Delta_{2i}

We consider α>0𝛼0\alpha>0 and 1/2<α012𝛼0-1/2<\alpha\leq 0 separately in G.2.1 and G.2.2, respectively.

G.2.1 Δ2isubscriptΔ2𝑖\Delta_{2i} under Assumption A.2 with α>0𝛼0\alpha>0

By the Cauchy-Schwarz inequality, we have

η(2p+n)/21F()θπ¯()dθdη2η(2p+n)/222()f()θπ¯()/π¯()2π¯()dθdη×η(2p+n)/2f()π¯()dθdη.superscriptdelimited-∥∥double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋d𝜃d𝜂\begin{split}&\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}\\ &\leq\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\|\nabla_{\theta}\bar{\pi}(\bullet)/\bar{\pi}(\bullet)\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta.\end{split} (G.5)

Similarly, by the Cauchy-Schwarz inequality, we have

η(2p+n)/21F()θπ¯()hi2()dθdη2η(2p+n)/222()f()θπ¯()/π¯()2π¯()hi2()dθdη×η(2p+n)/2f()π¯()hi2()dθdηη(2p+n)/222()f()θπ¯()/π¯()2π¯()dθdη×η(2p+n)/2f()π¯()hi2()dθdη,superscriptdelimited-∥∥double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋superscriptsubscript𝑖2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋superscriptsubscript𝑖2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋superscriptsubscript𝑖2d𝜃d𝜂double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋superscriptsubscript𝑖2d𝜃d𝜂\begin{split}&\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}\\ &\leq\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\|\nabla_{\theta}\bar{\pi}(\bullet)/\bar{\pi}(\bullet)\|^{2}\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\leq\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\|\nabla_{\theta}\bar{\pi}(\bullet)/\bar{\pi}(\bullet)\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta,\end{split} (G.6)

where the second inequality follows from the fact hi21superscriptsubscript𝑖21h_{i}^{2}\leq 1. Hence, by (G.5) and (G.6) with (G.2),

supiΔ2i2η(2p+n)/222()f()θπ¯()π¯()2π¯()dθdη.subscriptsupremum𝑖subscriptΔ2𝑖2double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓superscriptnormsubscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂\displaystyle\sup_{i}\Delta_{2i}\leq 2\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta.

By the relationship

θπ¯(ηθ2)2=4η2θ2{π¯(ηθ2)}2,superscriptnormsubscript𝜃¯𝜋𝜂superscriptnorm𝜃224superscript𝜂2superscriptnorm𝜃2superscriptsuperscript¯𝜋𝜂superscriptnorm𝜃22\|\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\|^{2}=4\eta^{2}\|\theta\|^{2}\{\bar{\pi}^{\prime}(\eta\|\theta\|^{2})\}^{2}, (G.7)

we have

supiΔ2i2subscriptsupremum𝑖subscriptΔ2𝑖2\displaystyle\frac{\sup_{i}\Delta_{2i}}{2} =η(2p+n)/222()f()θπ¯()π¯()2π¯()dθdηabsentdouble-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓superscriptnormsubscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂\displaystyle=\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
=4η(2p+n)/212(η{zθ2+1})f(η{zθ2+1})absent4double-integralsuperscript𝜂2𝑝𝑛21superscript2𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21\displaystyle=4\iint\eta^{(2p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})
×ηθ2{π¯(ηθ2)π¯(ηθ2)}2π¯(ηθ2)dθdηabsent𝜂superscriptnorm𝜃2superscriptsuperscript¯𝜋𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃22¯𝜋𝜂superscriptnorm𝜃2d𝜃d𝜂\displaystyle\qquad\times\eta\|\theta\|^{2}\left\{\frac{\bar{\pi}^{\prime}(\eta\|\theta\|^{2})}{\bar{\pi}(\eta\|\theta\|^{2})}\right\}^{2}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
4Π2η(2p+n)/212(η{zθ2+1})absent4superscriptΠ2double-integralsuperscript𝜂2𝑝𝑛21superscript2𝜂superscriptnorm𝑧𝜃21\displaystyle\leq 4\Pi^{2}\iint\eta^{(2p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z-\theta\|^{2}+1\})\
×f(η{zθ2+1})π¯(ηθ2)ηθ2dθdη,absent𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2d𝜃d𝜂\displaystyle\qquad\times f(\eta\{\|z-\theta\|^{2}+1\})\frac{\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta, (G.8)

where

Π=maxλ+λ|π¯(λ)|π¯(λ)=maxλ+|1p2+λπ(λ)π(λ)|,Πsubscript𝜆subscript𝜆superscript¯𝜋𝜆¯𝜋𝜆subscript𝜆subscript1𝑝2𝜆superscript𝜋𝜆𝜋𝜆\Pi=\max_{\lambda\in\mathbb{R}_{+}}\frac{\lambda|\bar{\pi}^{\prime}(\lambda)|}{\bar{\pi}(\lambda)}=\max_{\lambda\in\mathbb{R}_{+}}\left|1-\frac{p}{2}+\frac{\lambda\pi^{\prime}(\lambda)}{\pi(\lambda)}\right|, (G.9)

is bounded by Part 1 of Lemma E.1. Further, by (G.8), we have

18Π2supiΔ2idzη(p+n)/212(η{z2+1})f(η{z2+1})π¯(μ2)μ2dμdηdz=cpA3wp/21η(p+n)/212(η{w+1})f(η{w+1})dηdw=cpA30wp/21dw(1+w)(p+n)/20t(p+n)/212(t)f(t)dt=cpA2A3B(p/2,n/2),18superscriptΠ2subscriptsupremum𝑖subscriptΔ2𝑖d𝑧triple-integralsuperscript𝜂𝑝𝑛21superscript2𝜂superscriptdelimited-∥∥𝑧21𝑓𝜂superscriptdelimited-∥∥𝑧21¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇differential-d𝜂differential-d𝑧subscript𝑐𝑝subscript𝐴3double-integralsuperscript𝑤𝑝21superscript𝜂𝑝𝑛21superscript2𝜂𝑤1𝑓𝜂𝑤1differential-d𝜂differential-d𝑤subscript𝑐𝑝subscript𝐴3superscriptsubscript0superscript𝑤𝑝21d𝑤superscript1𝑤𝑝𝑛2superscriptsubscript0superscript𝑡𝑝𝑛21superscript2𝑡𝑓𝑡differential-d𝑡subscript𝑐𝑝subscript𝐴2subscript𝐴3𝐵𝑝2𝑛2\begin{split}&\frac{1}{8\Pi^{2}}\int\sup_{i}\Delta_{2i}\mathrm{d}z\\ &\leq\iiint\eta^{(p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z\|^{2}+1\})f(\eta\{\|z\|^{2}+1\})\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu\mathrm{d}\eta\mathrm{d}z\\ &=c_{p}A_{3}\iint w^{p/2-1}\eta^{(p+n)/2-1}\mathcal{F}^{2}(\eta\{w+1\})f(\eta\{w+1\})\mathrm{d}\eta\mathrm{d}w\\ &=c_{p}A_{3}\int_{0}^{\infty}\frac{w^{p/2-1}\mathrm{d}w}{(1+w)^{(p+n)/2}}\int_{0}^{\infty}t^{(p+n)/2-1}\mathcal{F}^{2}(t)f(t)\mathrm{d}t\\ &=c_{p}A_{2}A_{3}B(p/2,n/2),\end{split} (G.10)

where A2subscript𝐴2A_{2} given by (G.4) is bounded and

A3=pπ¯(μ2)μ2dμ=0π(λ)λdλsubscript𝐴3subscriptsuperscript𝑝¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇superscriptsubscript0𝜋𝜆𝜆differential-d𝜆\displaystyle A_{3}=\int_{\mathbb{R}^{p}}\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu=\int_{0}^{\infty}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda (G.11)

is also bounded as shown in Parts 2 and 4 of Lemma E.1.

G.2.2 Δ2isubscriptΔ2𝑖\Delta_{2i} under Assumption A.2 with 1/2<α012𝛼0-1/2<\alpha\leq 0

Let

k(λ)=λ1/2I[0,1](λ)+I(1,)(λ).𝑘𝜆superscript𝜆12subscript𝐼01𝜆subscript𝐼1𝜆\displaystyle k(\lambda)=\lambda^{1/2}I_{[0,1]}(\lambda)+I_{(1,\infty)}(\lambda).

Note that 0k(λ)10𝑘𝜆10\leq k(\lambda)\leq 1 and 1/k(λ)11𝑘𝜆11/k(\lambda)\geq 1 for any λ0𝜆0\lambda\geq 0. Then, by Cauchy-Schwarz inequality, we have

η(2p+n)/21F()θπ¯()dθdη2η(2p+n)/222()f()k()θπ¯()π¯()2π¯()dθdη×η(2p+n)/2f()π¯()k()dθdη𝒥1(f,π)η(2p+n)/222()f()k()θπ¯()π¯()2π¯()dθdη×η(2p+n)/2f()π¯()dθdη,superscriptdelimited-∥∥double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓𝑘superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋𝑘d𝜃d𝜂subscript𝒥1𝑓𝜋double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓𝑘superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋d𝜃d𝜂\begin{split}&\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}\\ &\leq\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)k(\bullet)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\frac{\bar{\pi}(\bullet)}{k(\bullet)}\mathrm{d}\theta\mathrm{d}\eta\\ &\leq\mathcal{J}_{1}(f,\pi)\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)k(\bullet)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta,\end{split} (G.12)

where the second inequality with the constant 𝒥1(f,π)subscript𝒥1𝑓𝜋\mathcal{J}_{1}(f,\pi) follows from Lemma I.2, provided below in Appendix I. Similarly, by the Cauchy-Schwarz inequality, we have

η(2p+n)/21F()θπ¯()hi2()dθdη2η(2p+n)/222()f()k()θπ¯()π¯()2π¯()hi2()dθdη×η(2p+n)/2f()π¯()k()hi2()dθdη𝒥2(f,π)η(2p+n)/222()f()k()θπ¯()π¯()2π¯()dθdη×η(2p+n)/2f()π¯i()dθdη,superscriptdelimited-∥∥double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋superscriptsubscript𝑖2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓𝑘superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋superscriptsubscript𝑖2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋𝑘superscriptsubscript𝑖2d𝜃d𝜂subscript𝒥2𝑓𝜋double-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓𝑘superscriptdelimited-∥∥subscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖d𝜃d𝜂\begin{split}&\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}\\ &\leq\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)k(\bullet)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\frac{\bar{\pi}(\bullet)}{k(\bullet)}h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\leq\mathcal{J}_{2}(f,\pi)\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)k(\bullet)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta,\end{split} (G.13)

where the second inequality with the constant 𝒥2(f,π)subscript𝒥2𝑓𝜋\mathcal{J}_{2}(f,\pi) follows from Lemma I.2, provided below in Appendix I, and from the fact hi21superscriptsubscript𝑖21h_{i}^{2}\leq 1. Hence, by (G.7), (G.12), (G.13) and with ΠΠ\Pi given by (G.9), we have

supiΔ2i𝒥1(f,π)+𝒥2(f,π)subscriptsupremum𝑖subscriptΔ2𝑖subscript𝒥1𝑓𝜋subscript𝒥2𝑓𝜋\displaystyle\frac{\sup_{i}\Delta_{2i}}{\mathcal{J}_{1}(f,\pi)+\mathcal{J}_{2}(f,\pi)} η(2p+n)/222()f()k()θπ¯()π¯()2π¯()dθdηabsentdouble-integralsuperscript𝜂2𝑝𝑛22superscript2𝑓𝑘superscriptnormsubscript𝜃¯𝜋¯𝜋2¯𝜋differential-d𝜃differential-d𝜂\displaystyle\leq\iint\eta^{(2p+n)/2-2}\mathcal{F}^{2}(\circ)f(\circ)k(\bullet)\left\|\frac{\nabla_{\theta}\bar{\pi}(\bullet)}{\bar{\pi}(\bullet)}\right\|^{2}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
4Π2η(2p+n)/212()f()k()π¯()ηθ2dθdη.absent4superscriptΠ2double-integralsuperscript𝜂2𝑝𝑛21superscript2𝑓𝑘¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq 4\Pi^{2}\iint\eta^{(2p+n)/2-1}\mathcal{F}^{2}(\circ)f(\circ)k(\bullet)\frac{\bar{\pi}(\bullet)}{\eta\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta.

Therefore we have

14Π2{𝒥1(f,π)+𝒥2(f,π)}supiΔ2idz14superscriptΠ2subscript𝒥1𝑓𝜋subscript𝒥2𝑓𝜋subscriptsupremum𝑖subscriptΔ2𝑖d𝑧\displaystyle\frac{1}{4\Pi^{2}\left\{\mathcal{J}_{1}(f,\pi)+\mathcal{J}_{2}(f,\pi)\right\}}\int\sup_{i}\Delta_{2i}\mathrm{d}z
η(p+n)/212(η{z2+1})f(η{z2+1})k(μ2)π¯(μ2)μ2dμdηdzabsenttriple-integralsuperscript𝜂𝑝𝑛21superscript2𝜂superscriptnorm𝑧21𝑓𝜂superscriptnorm𝑧21𝑘superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇differential-d𝜂differential-d𝑧\displaystyle\leq\iiint\eta^{(p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z\|^{2}+1\})f(\eta\{\|z\|^{2}+1\})k(\|\mu\|^{2})\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu\mathrm{d}\eta\mathrm{d}z
=A4cpwp/21η(p+n)/212(η{w+1})f(η{w+1})dηdwabsentsubscript𝐴4subscript𝑐𝑝double-integralsuperscript𝑤𝑝21superscript𝜂𝑝𝑛21superscript2𝜂𝑤1𝑓𝜂𝑤1differential-d𝜂differential-d𝑤\displaystyle=A_{4}c_{p}\iint w^{p/2-1}\eta^{(p+n)/2-1}\mathcal{F}^{2}(\eta\{w+1\})f(\eta\{w+1\})\mathrm{d}\eta\mathrm{d}w (G.14)
=A4cpwp/21dw(1+w)(p+n)/20t(p+n)/212(t)f(t)dtabsentsubscript𝐴4subscript𝑐𝑝superscript𝑤𝑝21d𝑤superscript1𝑤𝑝𝑛2superscriptsubscript0superscript𝑡𝑝𝑛21superscript2𝑡𝑓𝑡differential-d𝑡\displaystyle=A_{4}c_{p}\int\frac{w^{p/2-1}\mathrm{d}w}{(1+w)^{(p+n)/2}}\int_{0}^{\infty}t^{(p+n)/2-1}\mathcal{F}^{2}(t)f(t)\mathrm{d}t
=A2A4cpB(p/2,n/2),absentsubscript𝐴2subscript𝐴4subscript𝑐𝑝𝐵𝑝2𝑛2\displaystyle=A_{2}A_{4}c_{p}B(p/2,n/2),

where A2subscript𝐴2A_{2} given by (G.4) is bounded and

A4subscript𝐴4\displaystyle A_{4} =pk(μ2)π¯(μ2)μ2dμabsentsubscriptsuperscript𝑝𝑘superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle=\int_{\mathbb{R}^{p}}k(\|\mu\|^{2})\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu
=μ1π¯(μ2)μdμ+μ>1π¯(μ2)μ2dμabsentsubscriptnorm𝜇1¯𝜋superscriptnorm𝜇2norm𝜇differential-d𝜇subscriptnorm𝜇1¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle=\int_{\|\mu\|\leq 1}\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|}\mathrm{d}\mu+\int_{\|\mu\|>1}\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu
=01π(λ)λ1/2dλ+1π(λ)λdλabsentsuperscriptsubscript01𝜋𝜆superscript𝜆12differential-d𝜆superscriptsubscript1𝜋𝜆𝜆differential-d𝜆\displaystyle=\int_{0}^{1}\frac{\pi(\lambda)}{\lambda^{1/2}}\mathrm{d}\lambda+\int_{1}^{\infty}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda

is bounded as shown in Parts 2 and 4 of Lemma E.1.

The proof of Theorem 4.2 Case I is thus completed by applying the dominated convergence theorem to diffB(δπ,δπi;πi)diff𝐵subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\mathrm{diff}B(\delta_{\pi},\delta_{\pi i};\pi_{i}) as noted above.

Appendix H Proof for case II

Recall, under Assumption A.3A.3.2, π𝜋\pi and π¯¯𝜋\bar{\pi} satisfy

λπ(λ)π(λ)=κ(λ), and λπ¯(λ)π¯(λ)=1p2+κ(λ),formulae-sequence𝜆superscript𝜋𝜆𝜋𝜆𝜅𝜆 and 𝜆superscript¯𝜋𝜆¯𝜋𝜆1𝑝2𝜅𝜆\displaystyle\lambda\frac{\pi^{\prime}(\lambda)}{\pi(\lambda)}=\kappa(\lambda),\text{ and }\lambda\frac{\bar{\pi}^{\prime}(\lambda)}{\bar{\pi}(\lambda)}=1-\frac{p}{2}+\kappa(\lambda),

where κ(λ)0𝜅𝜆0\kappa(\lambda)\to 0 as λ𝜆\lambda\to\infty.

With κ(λ)𝜅𝜆\kappa(\lambda), we have

θπ¯(ηθ2)=2ηθπ¯(ηθ2)=2ηθ{(1p/2)π¯(ηθ2)ηθ2+π¯(ηθ2)ηθ2κ(ηθ2)}.subscript𝜃¯𝜋𝜂superscriptdelimited-∥∥𝜃22𝜂𝜃superscript¯𝜋𝜂superscriptdelimited-∥∥𝜃22𝜂𝜃1𝑝2¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2𝜅𝜂superscriptdelimited-∥∥𝜃2\begin{split}\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})&=2\eta\theta\bar{\pi}^{\prime}(\eta\|\theta\|^{2})\\ &=2\eta\theta\left\{(1-p/2)\frac{\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}+\frac{\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}\kappa(\eta\|\theta\|^{2})\right\}.\end{split} (H.1)

By Lemma I.3 and the relationship (H.1), the integral included in (F.3) is rewritten as

zTη(2p+n)/21F()θπ¯()dθdηsuperscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂\displaystyle-z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
=(p2n+2n+pn+2)zTη(2p+n)/21F()θπ¯()dθdηabsent𝑝2𝑛2𝑛𝑝𝑛2superscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad=\left(\frac{p-2}{n+2}-\frac{n+p}{n+2}\right)z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
=p2n+2η(2p+n)/2f()π¯()dθdηabsent𝑝2𝑛2double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad=\frac{p-2}{n+2}\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
(n+p)(p2)n+2η(2p+n)/21F()π¯()dθdη𝑛𝑝𝑝2𝑛2double-integralsuperscript𝜂2𝑝𝑛21𝐹¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad\quad-\frac{(n+p)(p-2)}{n+2}\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
n+pn+2zTη(2p+n)/21F()θπ¯()dθdη𝑛𝑝𝑛2superscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad\quad-\frac{n+p}{n+2}z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
=p2n+2η(2p+n)/2f()π¯()dθdηabsent𝑝2𝑛2double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad=\frac{p-2}{n+2}\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
+(n+p)(p2)n+2zTθθ2θ2η(2p+n)/21F()π¯()dθdη𝑛𝑝𝑝2𝑛2double-integralsuperscript𝑧T𝜃superscriptnorm𝜃2superscriptnorm𝜃2superscript𝜂2𝑝𝑛21𝐹¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad\quad+\frac{(n+p)(p-2)}{n+2}\iint\frac{z^{\mathrm{\scriptscriptstyle T}}\theta-\|\theta\|^{2}}{\|\theta\|^{2}}\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
2n+pn+2zTθη(2p+n)/21F()κ()π¯()θ2dθdη2𝑛𝑝𝑛2superscript𝑧Tdouble-integral𝜃superscript𝜂2𝑝𝑛21𝐹𝜅¯𝜋superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\quad\quad-2\frac{n+p}{n+2}z^{\mathrm{\scriptscriptstyle T}}\iint\theta\eta^{(2p+n)/2-1}F(\circ)\frac{\kappa(\bullet)\bar{\pi}(\bullet)}{\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta

where, again, with the notation

=ηθ2,=η(zθ2+1).\displaystyle\bullet=\eta\|\theta\|^{2},\quad\circ=\eta(\|z-\theta\|^{2}+1).

Similarly, by Lemma I.3 and the relationship (H.1), the integral included in (F.3) is rewritten as

zTη(2p+n)/21F()θπ¯i()dθdηsuperscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle-z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
=p2n+2η(2p+n)/2f()π¯i()dθdηabsent𝑝2𝑛2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad=\frac{p-2}{n+2}\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
(n+p)(p2)n+2η(2p+n)/21F()π¯i()dθdη𝑛𝑝𝑝2𝑛2double-integralsuperscript𝜂2𝑝𝑛21𝐹subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad\quad-\frac{(n+p)(p-2)}{n+2}\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
n+pn+2zTη(2p+n)/21F()θπ¯i()dθdη𝑛𝑝𝑛2superscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹subscript𝜃subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad\quad-\frac{n+p}{n+2}z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\circ)\nabla_{\theta}\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
=p2n+2η(2p+n)/2f()π¯i()dθdηabsent𝑝2𝑛2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad=\frac{p-2}{n+2}\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
+(n+p)(p2)n+2zTθθ2θ2η(2p+n)/21F()π¯i()dθdη𝑛𝑝𝑝2𝑛2double-integralsuperscript𝑧T𝜃superscriptnorm𝜃2superscriptnorm𝜃2superscript𝜂2𝑝𝑛21𝐹subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad\quad+\frac{(n+p)(p-2)}{n+2}\iint\frac{z^{\mathrm{\scriptscriptstyle T}}\theta-\|\theta\|^{2}}{\|\theta\|^{2}}\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta
2n+pn+2zTθη(2p+n)/21F()κ()π¯i()θ2dθdη2𝑛𝑝𝑛2superscript𝑧Tdouble-integral𝜃superscript𝜂2𝑝𝑛21𝐹𝜅subscript¯𝜋𝑖superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\quad\quad-2\frac{n+p}{n+2}z^{\mathrm{\scriptscriptstyle T}}\iint\theta\eta^{(2p+n)/2-1}F(\circ)\frac{\kappa(\bullet)\bar{\pi}_{i}(\bullet)}{\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta
n+pn+2zTη(2p+n)/21F()π¯()θhi2()dθdη.𝑛𝑝𝑛2superscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹¯𝜋subscript𝜃superscriptsubscript𝑖2differential-d𝜃differential-d𝜂\displaystyle\quad\quad-\frac{n+p}{n+2}z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\nabla_{\theta}h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta.

Then diffB¯(z;δπ,δπi;πi)¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i}) given by (F.3) is rewritten as

diffB¯(z;δπ,δπi;πi)¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖\displaystyle\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})
=cnz2(n+p)2(n+2)2{zTη(2p+n)/21F()π¯()θhi2()dθdηη(2p+n)/2f()π¯i()dθdη\displaystyle=\frac{c_{n}}{\|z\|^{2}}\frac{(n+p)^{2}}{(n+2)^{2}}\left\{z^{\mathrm{\scriptscriptstyle T}}\frac{\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\nabla_{\theta}h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right.
+(p2)(zTθ/θ21)η(2p+n)/21F()π¯()dθdηη(2p+n)/2f()π¯()dθdη𝑝2double-integralsuperscript𝑧T𝜃superscriptnorm𝜃21superscript𝜂2𝑝𝑛21𝐹¯𝜋differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad\left.+(p-2)\frac{\iint(z^{\mathrm{\scriptscriptstyle T}}\theta/\|\theta\|^{2}-1)\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right.
(p2)(zTθ/θ21)η(2p+n)/21F()π¯i()dθdηη(2p+n)/2f()π¯i()dθdη𝑝2double-integralsuperscript𝑧T𝜃superscriptnorm𝜃21superscript𝜂2𝑝𝑛21𝐹subscript¯𝜋𝑖differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad\left.-(p-2)\frac{\iint(z^{\mathrm{\scriptscriptstyle T}}\theta/\|\theta\|^{2}-1)\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right.
2zTθη(2p+n)/21F(){κ()π¯()/θ2}dθdηη(2p+n)/2f()π¯()dθdη2superscript𝑧Tdouble-integral𝜃superscript𝜂2𝑝𝑛21𝐹𝜅¯𝜋superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle\quad\left.-2\frac{z^{\mathrm{\scriptscriptstyle T}}\iint\theta\eta^{(2p+n)/2-1}F(\circ)\{\kappa(\bullet)\bar{\pi}(\bullet)/\|\theta\|^{2}\}\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right.
+2zTθη(2p+n)/21F(){κ()π¯i()/θ2}dθdηη(2p+n)/2f()π¯i()dθdη}2\displaystyle\quad\left.+2\frac{z^{\mathrm{\scriptscriptstyle T}}\iint\theta\eta^{(2p+n)/2-1}F(\circ)\{\kappa(\bullet)\bar{\pi}_{i}(\bullet)/\|\theta\|^{2}\}\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}\right\}^{2}
×η(2p+n)/2f()π¯i()dθdη.\displaystyle\qquad\times\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta.

By the triangle inequality and the fact hi21superscriptsubscript𝑖21h_{i}^{2}\leq 1,

diffB¯(z;δπ,δπi;πi)2cn(n+p)2(n+2)2{Δ1i+(p2)2Δ3i+4Δ4i},¯diff𝐵𝑧subscript𝛿𝜋subscript𝛿𝜋𝑖subscript𝜋𝑖2subscript𝑐𝑛superscript𝑛𝑝2superscript𝑛22subscriptΔ1𝑖superscript𝑝22subscriptΔ3𝑖4subscriptΔ4𝑖\displaystyle\overline{\mathrm{diff}B}(z;\delta_{\pi},\delta_{\pi i};\pi_{i})\leq 2\frac{c_{n}(n+p)^{2}}{(n+2)^{2}}\left\{\Delta_{1i}+(p-2)^{2}\Delta_{3i}+4\Delta_{4i}\right\},

where

Δ1isubscriptΔ1𝑖\displaystyle\Delta_{1i} =η(2p+n)/21F()π¯()θhi2()dθdη2η(2p+n)/2f()π¯i()dθdη,absentsuperscriptnormdouble-integralsuperscript𝜂2𝑝𝑛21𝐹¯𝜋subscript𝜃superscriptsubscript𝑖2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle=\frac{\left\|\iint\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\nabla_{\theta}h_{i}^{2}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta},
Δ3isubscriptΔ3𝑖\displaystyle\Delta_{3i} =1z2{(zTθ/θ21)η(2p+n)/21F()π¯()dθdη}2η(2p+n)/2f()π¯()dθdηabsent1superscriptnorm𝑧2superscriptdouble-integralsuperscript𝑧T𝜃superscriptnorm𝜃21superscript𝜂2𝑝𝑛21𝐹¯𝜋differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle=\frac{1}{\|z\|^{2}}\frac{\left\{\iint(z^{\mathrm{\scriptscriptstyle T}}\theta/\|\theta\|^{2}-1)\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\}^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}
+1z2{(zTθ/θ21)η(2p+n)/21F()π¯i()dθdη}2η(2p+n)/2f()π¯i()dθdη,1superscriptnorm𝑧2superscriptdouble-integralsuperscript𝑧T𝜃superscriptnorm𝜃21superscript𝜂2𝑝𝑛21𝐹subscript¯𝜋𝑖differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad+\frac{1}{\|z\|^{2}}\frac{\left\{\iint(z^{\mathrm{\scriptscriptstyle T}}\theta/\|\theta\|^{2}-1)\eta^{(2p+n)/2-1}F(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta\right\}^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}, (H.2)
Δ4isubscriptΔ4𝑖\displaystyle\Delta_{4i} =θη(2p+n)/21F()κ()π¯()θ2dθdη2η(2p+n)/2f()π¯()dθdηabsentsuperscriptnormdouble-integral𝜃superscript𝜂2𝑝𝑛21𝐹𝜅¯𝜋superscriptnorm𝜃2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓¯𝜋differential-d𝜃differential-d𝜂\displaystyle=\frac{\left\|\iint\theta\eta^{(2p+n)/2-1}F(\circ)\kappa(\bullet)\bar{\pi}(\bullet)\|\theta\|^{-2}\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}
+θη(2p+n)/21F()κ()π¯i()θ2dθdη2η(2p+n)/2f()π¯i()dθdη.superscriptnormdouble-integral𝜃superscript𝜂2𝑝𝑛21𝐹𝜅subscript¯𝜋𝑖superscriptnorm𝜃2differential-d𝜃differential-d𝜂2double-integralsuperscript𝜂2𝑝𝑛2𝑓subscript¯𝜋𝑖differential-d𝜃differential-d𝜂\displaystyle\quad+\frac{\left\|\iint\theta\eta^{(2p+n)/2-1}F(\circ)\kappa(\bullet)\bar{\pi}_{i}(\bullet)\|\theta\|^{-2}\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\iint\eta^{(2p+n)/2}f(\circ)\bar{\pi}_{i}(\bullet)\mathrm{d}\theta\mathrm{d}\eta}. (H.3)

For Δ1isubscriptΔ1𝑖\Delta_{1i}, as seen in Section G.1, we have supiΔ1idz<subscriptsupremum𝑖subscriptΔ1𝑖d𝑧\int\sup_{i}\Delta_{1i}\mathrm{d}z<\infty. We will show integrability supiΔ3idz<subscriptsupremum𝑖subscriptΔ3𝑖d𝑧\int\sup_{i}\Delta_{3i}\mathrm{d}z<\infty and integrability supiΔ4idz<subscriptsupremum𝑖subscriptΔ4𝑖d𝑧\int\sup_{i}\Delta_{4i}\mathrm{d}z<\infty in Sub-sections H.1 and H.2, respectively.

H.1 Δ3isubscriptΔ3𝑖\Delta_{3i}

Note the inequality

|zTθθ21|=|(zθ)Tθθ2|zθθzθ2+1θ.superscript𝑧T𝜃superscriptnorm𝜃21superscript𝑧𝜃T𝜃superscriptnorm𝜃2norm𝑧𝜃norm𝜃superscriptnorm𝑧𝜃21norm𝜃\displaystyle\left|\frac{z^{\mathrm{\scriptscriptstyle T}}\theta}{\|\theta\|^{2}}-1\right|=\left|\frac{(z-\theta)^{\mathrm{\scriptscriptstyle T}}\theta}{\|\theta\|^{2}}\right|\leq\frac{\|z-\theta\|}{\|\theta\|}\leq\frac{\sqrt{\|z-\theta\|^{2}+1}}{\|\theta\|}.

Then, in the first and second terms of (H.2), we have

|zTθθ2θ2|η(2p+n)/21F(η{zθ2+1})π¯(ηθ2)dθdηdouble-integralsuperscript𝑧T𝜃superscriptnorm𝜃2superscriptnorm𝜃2superscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\iint\left|\frac{z^{\mathrm{\scriptscriptstyle T}}\theta-\|\theta\|^{2}}{\|\theta\|^{2}}\right|\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
η(2p+n)/21η1/2θ{η(zθ2+1)}1/2F(η{zθ2+1})π¯(ηθ2)dθdηabsentdouble-integralsuperscript𝜂2𝑝𝑛21superscript𝜂12norm𝜃superscript𝜂superscriptnorm𝑧𝜃2112𝐹𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq\iint\frac{\eta^{(2p+n)/2-1}}{\eta^{1/2}\|\theta\|}\{\eta(\|z-\theta\|^{2}+1)\}^{1/2}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
=η(2p+n)/21η1/2θ~(η{zθ2+1})f(η{zθ2+1})π¯(ηθ2)dθdηabsentdouble-integralsuperscript𝜂2𝑝𝑛21superscript𝜂12norm𝜃~𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\iint\frac{\eta^{(2p+n)/2-1}}{\eta^{1/2}\|\theta\|}\tilde{\mathcal{F}}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta

where

~(t)=t1/2(t)=t1/2F(t)f(t)~𝑡superscript𝑡12𝑡superscript𝑡12𝐹𝑡𝑓𝑡\displaystyle\tilde{\mathcal{F}}(t)=t^{1/2}\mathcal{F}(t)=t^{1/2}\frac{F(t)}{f(t)}

and

|zTθθ2θ2|η(2p+n)/21F(η{zθ2+1})π¯i(ηθ2)dθdηdouble-integralsuperscript𝑧T𝜃superscriptnorm𝜃2superscriptnorm𝜃2superscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21subscript¯𝜋𝑖𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\iint\left|\frac{z^{\mathrm{\scriptscriptstyle T}}\theta-\|\theta\|^{2}}{\|\theta\|^{2}}\right|\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}_{i}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
η(2p+n)/21η1/2θ~(η{zθ2+1})f(η{zθ2+1})π¯i(ηθ2)dθdη.absentdouble-integralsuperscript𝜂2𝑝𝑛21superscript𝜂12norm𝜃~𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21subscript¯𝜋𝑖𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq\iint\frac{\eta^{(2p+n)/2-1}}{\eta^{1/2}\|\theta\|}\tilde{\mathcal{F}}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}_{i}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta.

Under Assumption A.2 on π𝜋\pi with α>0𝛼0\alpha>0, applying the same technique used in Sub-Section G.2.1, the integrability of

B1subscript𝐵1\displaystyle B_{1} =η(2p+n)/22z2~2(η{zθ2+1})f(η{zθ2+1})absenttriple-integralsuperscript𝜂2𝑝𝑛22superscriptnorm𝑧2superscript~2𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21\displaystyle=\iiint\frac{\eta^{(2p+n)/2-2}}{\|z\|^{2}}\tilde{\mathcal{F}}^{2}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})
×π¯(ηθ2)ηθ2dθdηdzabsent¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2d𝜃d𝜂d𝑧\displaystyle\qquad\times\frac{\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta\mathrm{d}z

implies the integrability of supiΔ3idzsubscriptsupremum𝑖subscriptΔ3𝑖d𝑧\int\sup_{i}\Delta_{3i}\mathrm{d}z. The integrability of B1subscript𝐵1B_{1} is shown as follows;

B1subscript𝐵1\displaystyle B_{1} =ηn/21y2~2(yμ2+η)f(yμ2+η)π¯(μ2)μ2dμdηdyabsenttriple-integralsuperscript𝜂𝑛21superscriptnorm𝑦2superscript~2superscriptnorm𝑦𝜇2𝜂𝑓superscriptnorm𝑦𝜇2𝜂¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇differential-d𝜂differential-d𝑦\displaystyle=\iiint\frac{\eta^{n/2-1}}{\|y\|^{2}}\tilde{\mathcal{F}}^{2}(\|y-\mu\|^{2}+\eta)f(\|y-\mu\|^{2}+\eta)\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu\mathrm{d}\eta\mathrm{d}y
=p(pf(yμ2)y2dy)π¯(μ2)μ2dμabsentsubscriptsuperscript𝑝subscriptsuperscript𝑝subscript𝑓superscriptnorm𝑦𝜇2superscriptnorm𝑦2differential-d𝑦¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle=\int_{\mathbb{R}^{p}}\left(\int_{\mathbb{R}^{p}}\frac{f_{\star}(\|y-\mu\|^{2})}{\|y\|^{2}}\mathrm{d}y\right)\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu
𝒬fpmin(1,μ2)π¯(μ2)μ2dμabsentsubscript𝒬𝑓subscriptsuperscript𝑝1superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle\leq\mathcal{Q}_{f}\int_{\mathbb{R}^{p}}\min(1,\|\mu\|^{-2})\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu (H.4)
=𝒬f{μ1π¯(μ2)μ2dμ+μ>1π¯(μ2)μ4dμ}absentsubscript𝒬𝑓subscriptnorm𝜇1¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇subscriptnorm𝜇1¯𝜋superscriptnorm𝜇2superscriptnorm𝜇4differential-d𝜇\displaystyle=\mathcal{Q}_{f}\left\{\int_{\|\mu\|\leq 1}\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu+\int_{\|\mu\|>1}\frac{\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{4}}\mathrm{d}\mu\right\}
=𝒬f{01π(λ)λdλ+1π(λ)λ2dλ}<absentsubscript𝒬𝑓superscriptsubscript01𝜋𝜆𝜆differential-d𝜆superscriptsubscript1𝜋𝜆superscript𝜆2differential-d𝜆\displaystyle=\mathcal{Q}_{f}\left\{\int_{0}^{1}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\lambda+\int_{1}^{\infty}\frac{\pi(\lambda)}{\lambda^{2}}\mathrm{d}\lambda\right\}<\infty

where

f(t)=0ηn/21~2(t+η)f(t+η)dη,subscript𝑓𝑡superscriptsubscript0superscript𝜂𝑛21superscript~2𝑡𝜂𝑓𝑡𝜂differential-d𝜂\displaystyle f_{\star}(t)=\int_{0}^{\infty}\eta^{n/2-1}\tilde{\mathcal{F}}^{2}(t+\eta)f(t+\eta)\mathrm{d}\eta,

the inequality with 𝒬fsubscript𝒬𝑓\mathcal{Q}_{f} follows from Part 2 of Lemma E.3 and the integrability of the right-hand side follows from Parts 3 and 5 of Lemma E.1.

Under Assumption A.2 on π𝜋\pi with 1/2<α012𝛼0-1/2<\alpha\leq 0, applying the same technique used in Sub-Section G.2.2, the integrability of

B2subscript𝐵2\displaystyle B_{2} =η(2p+n)/22z2~2(η{zθ2+1})f(η{zθ2+1})absenttriple-integralsuperscript𝜂2𝑝𝑛22superscriptnorm𝑧2superscript~2𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21\displaystyle=\iiint\frac{\eta^{(2p+n)/2-2}}{\|z\|^{2}}\tilde{\mathcal{F}}^{2}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})
×k(ηθ2)π¯(ηθ2)ηθ2dθdηdzabsent𝑘𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2d𝜃d𝜂d𝑧\displaystyle\qquad\times\frac{k(\eta\|\theta\|^{2})\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta\mathrm{d}z

where k(λ)=λ1/2I[0,1](λ)+I(1,)(λ)𝑘𝜆superscript𝜆12subscript𝐼01𝜆subscript𝐼1𝜆k(\lambda)=\lambda^{1/2}I_{[0,1]}(\lambda)+I_{(1,\infty)}(\lambda), implies the integrability of supiΔ3idzsubscriptsupremum𝑖subscriptΔ3𝑖d𝑧\int\sup_{i}\Delta_{3i}\mathrm{d}z. As in (H.4), B2subscript𝐵2B_{2} is given by

B2subscript𝐵2\displaystyle B_{2} 𝒬fpmin(1,μ2)k(μ2)π¯(μ2)μ2dμabsentsubscript𝒬𝑓subscriptsuperscript𝑝1superscriptnorm𝜇2𝑘superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle\leq\mathcal{Q}_{f}\int_{\mathbb{R}^{p}}\min(1,\|\mu\|^{-2})\frac{k(\|\mu\|^{2})\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu
=𝒬f{01π(λ)λ1/2dλ+1π(λ)λ2dλ}<absentsubscript𝒬𝑓superscriptsubscript01𝜋𝜆superscript𝜆12differential-d𝜆superscriptsubscript1𝜋𝜆superscript𝜆2differential-d𝜆\displaystyle=\mathcal{Q}_{f}\left\{\int_{0}^{1}\frac{\pi(\lambda)}{\lambda^{1/2}}\mathrm{d}\lambda+\int_{1}^{\infty}\frac{\pi(\lambda)}{\lambda^{2}}\mathrm{d}\lambda\right\}<\infty

which is bounded by Parts 2 and 5 of Lemma E.1.

H.2 Δ4isubscriptΔ4𝑖\Delta_{4i}

Under Assumption A.2 on π𝜋\pi with α>0𝛼0\alpha>0, applying the same technique used in Sub-Section G.2.1, the integrability of

B3subscript𝐵3\displaystyle B_{3} =η(2p+n)/212(η{zθ2+1})f(η{zθ2+1})absenttriple-integralsuperscript𝜂2𝑝𝑛21superscript2𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21\displaystyle=\iiint\eta^{(2p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})
×κ2(ηθ2)π¯(ηθ2)ηθ2dθdηdzabsentsuperscript𝜅2𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2d𝜃d𝜂d𝑧\displaystyle\qquad\times\frac{\kappa^{2}(\eta\|\theta\|^{2})\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta\mathrm{d}z

implies the integrability of supiΔ4idzsubscriptsupremum𝑖subscriptΔ4𝑖d𝑧\int\sup_{i}\Delta_{4i}\mathrm{d}z. The integrability of B3subscript𝐵3B_{3} is shown as follows;

B3subscript𝐵3\displaystyle B_{3} =cpA2B(p/2,n/2)κ2(μ2)π¯(μ2)μ2dμabsentsubscript𝑐𝑝subscript𝐴2𝐵𝑝2𝑛2superscript𝜅2superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle=c_{p}A_{2}B(p/2,n/2)\int\frac{\kappa^{2}(\|\mu\|^{2})\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu
=cpA2B(p/2,n/2){supλ(0,1)κ2(λ)01π(λ)λdμ+1π(λ)κ2(λ)λdμ},absentsubscript𝑐𝑝subscript𝐴2𝐵𝑝2𝑛2subscriptsupremum𝜆01superscript𝜅2𝜆superscriptsubscript01𝜋𝜆𝜆differential-d𝜇superscriptsubscript1𝜋𝜆superscript𝜅2𝜆𝜆differential-d𝜇\displaystyle=c_{p}A_{2}B(p/2,n/2)\left\{\sup_{\lambda\in(0,1)}\kappa^{2}(\lambda)\int_{0}^{1}\frac{\pi(\lambda)}{\lambda}\mathrm{d}\mu+\int_{1}^{\infty}\frac{\pi(\lambda)\kappa^{2}(\lambda)}{\lambda}\mathrm{d}\mu\right\},

where A2subscript𝐴2A_{2} is given by (G.4), the first term is bounded by Parts 1 and 3 of Lemma E.1 and the second term is bounded by Part 8 of Lemma E.1.

Under Assumption A.2 on π𝜋\pi with 1/2<α012𝛼0-1/2<\alpha\leq 0, applying the same technique used in Sub-Section G.2.2 the integrability of

B4subscript𝐵4\displaystyle B_{4} =η(2p+n)/212(η{zθ2+1})f(η{zθ2+1})absenttriple-integralsuperscript𝜂2𝑝𝑛21superscript2𝜂superscriptnorm𝑧𝜃21𝑓𝜂superscriptnorm𝑧𝜃21\displaystyle=\iiint\eta^{(2p+n)/2-1}\mathcal{F}^{2}(\eta\{\|z-\theta\|^{2}+1\})f(\eta\{\|z-\theta\|^{2}+1\})
×k(ηθ2)κ2(ηθ2)π¯(ηθ2)ηθ2dθdηdzabsent𝑘𝜂superscriptnorm𝜃2superscript𝜅2𝜂superscriptnorm𝜃2¯𝜋𝜂superscriptnorm𝜃2𝜂superscriptnorm𝜃2d𝜃d𝜂d𝑧\displaystyle\qquad\times\frac{k(\eta\|\theta\|^{2})\kappa^{2}(\eta\|\theta\|^{2})\bar{\pi}(\eta\|\theta\|^{2})}{\eta\|\theta\|^{2}}\mathrm{d}\theta\mathrm{d}\eta\mathrm{d}z

where k(λ)=λ1/2I[0,1](λ)+I(1,)(λ)𝑘𝜆superscript𝜆12subscript𝐼01𝜆subscript𝐼1𝜆k(\lambda)=\lambda^{1/2}I_{[0,1]}(\lambda)+I_{(1,\infty)}(\lambda) implies the integrability of supiΔ4idzsubscriptsupremum𝑖subscriptΔ4𝑖d𝑧\int\sup_{i}\Delta_{4i}\mathrm{d}z. The integrability of B3subscript𝐵3B_{3} is shown as follows;

B4subscript𝐵4\displaystyle B_{4} =cpA2B(p/2,n/2)k(μ2)κ2(μ2)π¯(μ2)μ2dμabsentsubscript𝑐𝑝subscript𝐴2𝐵𝑝2𝑛2𝑘superscriptnorm𝜇2superscript𝜅2superscriptnorm𝜇2¯𝜋superscriptnorm𝜇2superscriptnorm𝜇2differential-d𝜇\displaystyle=c_{p}A_{2}B(p/2,n/2)\int\frac{k(\|\mu\|^{2})\kappa^{2}(\|\mu\|^{2})\bar{\pi}(\|\mu\|^{2})}{\|\mu\|^{2}}\mathrm{d}\mu
=cpA2B(p/2,n/2){supλ(0,1)κ2(λ)01π(λ)λ1/2dμ+1π(λ)κ2(λ)λdμ},absentsubscript𝑐𝑝subscript𝐴2𝐵𝑝2𝑛2subscriptsupremum𝜆01superscript𝜅2𝜆superscriptsubscript01𝜋𝜆superscript𝜆12differential-d𝜇superscriptsubscript1𝜋𝜆superscript𝜅2𝜆𝜆differential-d𝜇\displaystyle=c_{p}A_{2}B(p/2,n/2)\left\{\sup_{\lambda\in(0,1)}\kappa^{2}(\lambda)\int_{0}^{1}\frac{\pi(\lambda)}{\lambda^{1/2}}\mathrm{d}\mu+\int_{1}^{\infty}\frac{\pi(\lambda)\kappa^{2}(\lambda)}{\lambda}\mathrm{d}\mu\right\},

where A2subscript𝐴2A_{2} is given by (G.4), the first term is bounded by Parts 1 and 2 of Lemma E.1 and the second term is bounded by Part 8 of Lemma E.1.

Appendix I Additional Lemmas used in Sections G and H

Let

J(f,π,z)𝐽𝑓𝜋𝑧\displaystyle J(f,\pi,z) (I.1)
=ηθ21η(2p+n)/2{ηθ2}1/2π¯(ηθ2)f(η{zθ2+1})dθdηηθ21η(2p+n)/2π¯(ηθ2)f(η{zθ2+1})dθdη.absentsubscriptdouble-integral𝜂superscriptnorm𝜃21superscript𝜂2𝑝𝑛2superscript𝜂superscriptnorm𝜃212¯𝜋𝜂superscriptnorm𝜃2𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜃differential-d𝜂subscriptdouble-integral𝜂superscriptnorm𝜃21superscript𝜂2𝑝𝑛2¯𝜋𝜂superscriptnorm𝜃2𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜃differential-d𝜂\displaystyle=\frac{\iint_{\eta\|\theta\|^{2}\leq 1}\eta^{(2p+n)/2}\{\eta\|\theta\|^{2}\}^{-1/2}\bar{\pi}(\eta\|\theta\|^{2})f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\theta\mathrm{d}\eta}{\iint_{\eta\|\theta\|^{2}\leq 1}\eta^{(2p+n)/2}\bar{\pi}(\eta\|\theta\|^{2})f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\theta\mathrm{d}\eta}.

Then we have a following result.

Lemma I.1.

Suppose Assumptions F.1F.3 on f𝑓f hold. Assume Assumptions A.2 on π𝜋\pi with 1/2<α012𝛼0-1/2<\alpha\leq 0. Then

J(f,π,z)𝒥(f,π)<𝐽𝑓𝜋𝑧𝒥𝑓𝜋J(f,\pi,z)\leq\mathcal{J}(f,\pi)<\infty (I.2)

for any zp𝑧superscript𝑝z\in\mathbb{R}^{p}, where

𝒥(f,π)=2α+1α+1/2maxa+φ(a;α+1/2)mina+φ(a;α+1)maxλ[0,1]ν(λ)minλ[0,1]ν(λ),φ(a;γ)=0at(p+n)/2+γf(t)dt0at(p+n)/2+γfG(t)dt,fG(t)=(2π)(p+n)/2exp(t/2).formulae-sequence𝒥𝑓𝜋2𝛼1𝛼12subscript𝑎subscript𝜑𝑎𝛼12subscript𝑎subscript𝜑𝑎𝛼1subscript𝜆01𝜈𝜆subscript𝜆01𝜈𝜆formulae-sequence𝜑𝑎𝛾superscriptsubscript0𝑎superscript𝑡𝑝𝑛2𝛾𝑓𝑡differential-d𝑡superscriptsubscript0𝑎superscript𝑡𝑝𝑛2𝛾subscript𝑓𝐺𝑡differential-d𝑡subscript𝑓𝐺𝑡superscript2𝜋𝑝𝑛2𝑡2\begin{split}\mathcal{J}(f,\pi)&=2\frac{\alpha+1}{\alpha+1/2}\frac{\max_{a\in\mathbb{R}_{+}}\varphi(a;\alpha+1/2)}{\min_{a\in\mathbb{R}_{+}}\varphi(a;\alpha+1)}\frac{\max_{\lambda\in[0,1]}\nu(\lambda)}{\min_{\lambda\in[0,1]}\nu(\lambda)},\\ \varphi(a;\gamma)&=\frac{\int_{0}^{a}t^{(p+n)/2+\gamma}f(t)\mathrm{d}t}{\int_{0}^{a}t^{(p+n)/2+\gamma}f_{G}(t)\mathrm{d}t},\\ f_{G}(t)&=(2\pi)^{-(p+n)/2}\exp(-t/2).\end{split} (I.3)
Proof.

By Assumptions A.2 on π𝜋\pi,

J(f,π,z)maxλ[0,1]ν(λ)minλ[0,1]ν(λ)J1(f,π,z)𝐽𝑓𝜋𝑧subscript𝜆01𝜈𝜆subscript𝜆01𝜈𝜆subscript𝐽1𝑓𝜋𝑧J(f,\pi,z)\leq\frac{\max_{\lambda\in[0,1]}\nu(\lambda)}{\min_{\lambda\in[0,1]}\nu(\lambda)}J_{1}(f,\pi,z) (I.4)

where

J1(f,π,z)subscript𝐽1𝑓𝜋𝑧\displaystyle J_{1}(f,\pi,z) (I.5)
=ηθ21η(2p+n)/2{ηθ2}α+(1p)/2f(η{zθ2+1})dθdηηθ21η(2p+n)/2{ηθ2}α+(2p)/2f(η{zθ2+1})dθdηabsentsubscriptdouble-integral𝜂superscriptnorm𝜃21superscript𝜂2𝑝𝑛2superscript𝜂superscriptnorm𝜃2𝛼1𝑝2𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜃differential-d𝜂subscriptdouble-integral𝜂superscriptnorm𝜃21superscript𝜂2𝑝𝑛2superscript𝜂superscriptnorm𝜃2𝛼2𝑝2𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜃differential-d𝜂\displaystyle=\frac{\iint_{\eta\|\theta\|^{2}\leq 1}\eta^{(2p+n)/2}\{\eta\|\theta\|^{2}\}^{\alpha+(1-p)/2}f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\theta\mathrm{d}\eta}{\iint_{\eta\|\theta\|^{2}\leq 1}\eta^{(2p+n)/2}\{\eta\|\theta\|^{2}\}^{\alpha+(2-p)/2}f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\theta\mathrm{d}\eta}
=pθ2α+1p{01/θ2η(p+n+1)/2+αf(η{zθ2+1})dη}dθpθ2α+2p{01/θ2η(p+n+2)/2+αf(η{zθ2+1})dη}dθ.absentsubscriptsuperscript𝑝superscriptnorm𝜃2𝛼1𝑝superscriptsubscript01superscriptnorm𝜃2superscript𝜂𝑝𝑛12𝛼𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜂differential-d𝜃subscriptsuperscript𝑝superscriptnorm𝜃2𝛼2𝑝superscriptsubscript01superscriptnorm𝜃2superscript𝜂𝑝𝑛22𝛼𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜂differential-d𝜃\displaystyle=\frac{\int_{\mathbb{R}^{p}}\|\theta\|^{2\alpha+1-p}\left\{\int_{0}^{1/\|\theta\|^{2}}\eta^{(p+n+1)/2+\alpha}f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\eta\right\}\mathrm{d}\theta}{\int_{\mathbb{R}^{p}}\|\theta\|^{2\alpha+2-p}\left\{\int_{0}^{1/\|\theta\|^{2}}\eta^{(p+n+2)/2+\alpha}f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\eta\right\}\mathrm{d}\theta}.

Let γ=α+1𝛾𝛼1\gamma=\alpha+1 for the denominator and α+1/2𝛼12\alpha+1/2 for the numerator of J1(f,π,z)subscript𝐽1𝑓𝜋𝑧J_{1}(f,\pi,z). By change of variables, the integral with respect to η𝜂\eta is rewritten as

01/θ2η(p+n)/2+γf(η{zθ2+1})dηsuperscriptsubscript01superscriptnorm𝜃2superscript𝜂𝑝𝑛2𝛾𝑓𝜂superscriptnorm𝑧𝜃21differential-d𝜂\displaystyle\int_{0}^{1/\|\theta\|^{2}}\eta^{(p+n)/2+\gamma}f(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\eta
={zθ2+1}(p+n)/21γ0at(p+n)/2+γf(t)dtabsentsuperscriptsuperscriptnorm𝑧𝜃21𝑝𝑛21𝛾superscriptsubscript0𝑎superscript𝑡𝑝𝑛2𝛾𝑓𝑡differential-d𝑡\displaystyle=\{\|z-\theta\|^{2}+1\}^{-(p+n)/2-1-\gamma}\int_{0}^{a}t^{(p+n)/2+\gamma}f(t)\mathrm{d}t
=φ(a;γ)01/θ2η(p+n)/2+γfG(η{zθ2+1})dηabsent𝜑𝑎𝛾superscriptsubscript01superscriptnorm𝜃2superscript𝜂𝑝𝑛2𝛾subscript𝑓𝐺𝜂superscriptnorm𝑧𝜃21differential-d𝜂\displaystyle=\varphi(a;\gamma)\int_{0}^{1/\|\theta\|^{2}}\eta^{(p+n)/2+\gamma}f_{G}(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\eta

where φ(a;γ)𝜑𝑎𝛾\varphi(a;\gamma) is defined by (I.3) and a={zθ2+1}/θ2𝑎superscriptnorm𝑧𝜃21superscriptnorm𝜃2a=\{\|z-\theta\|^{2}+1\}/\|\theta\|^{2}. Note

lima0φ(a;γ)=f(0)fG(0) and limaφ(a;γ)=0t(p+n)/2+γf(t)dt0t(p+n)/2+γfG(t)dt,subscript𝑎0𝜑𝑎𝛾𝑓0subscript𝑓𝐺0 and subscript𝑎𝜑𝑎𝛾superscriptsubscript0superscript𝑡𝑝𝑛2𝛾𝑓𝑡differential-d𝑡superscriptsubscript0superscript𝑡𝑝𝑛2𝛾subscript𝑓𝐺𝑡differential-d𝑡\displaystyle\lim_{a\to 0}\varphi(a;\gamma)=\frac{f(0)}{f_{G}(0)}\text{ and }\lim_{a\to\infty}\varphi(a;\gamma)=\frac{\int_{0}^{\infty}t^{(p+n)/2+\gamma}f(t)\mathrm{d}t}{\int_{0}^{\infty}t^{(p+n)/2+\gamma}f_{G}(t)\mathrm{d}t},

which are both positive and bounded from the above under 0<γ10𝛾10<\gamma\leq 1 and under Assumptions F.1F.3 on f𝑓f and hence

mina+φ(a;γ)>0 and maxa+φ(a;γ)<,subscript𝑎subscript𝜑𝑎𝛾0 and subscript𝑎subscript𝜑𝑎𝛾\displaystyle\min_{a\in\mathbb{R}_{+}}\varphi(a;\gamma)>0\text{ and }\max_{a\in\mathbb{R}_{+}}\varphi(a;\gamma)<\infty,

under 0<γ10𝛾10<\gamma\leq 1. Therefore we have

J1(f,π,z)maxa+φ(a;α+1/2)mina+φ(a;α+1)J2(f,π,z)subscript𝐽1𝑓𝜋𝑧subscript𝑎subscript𝜑𝑎𝛼12subscript𝑎subscript𝜑𝑎𝛼1subscript𝐽2𝑓𝜋𝑧J_{1}(f,\pi,z)\leq\frac{\max_{a\in\mathbb{R}_{+}}\varphi(a;\alpha+1/2)}{\min_{a\in\mathbb{R}_{+}}\varphi(a;\alpha+1)}J_{2}(f,\pi,z) (I.6)

where

J2(f,π,z)subscript𝐽2𝑓𝜋𝑧\displaystyle J_{2}(f,\pi,z) (I.7)
=ηθ21η(2p+n)/2{ηθ2}α+(1p)/2fG(η{zθ2+1})dθdηηθ21η(2p+n)/2{ηθ2}α+(2p)/2fG(η{zθ2+1})dθdηabsentsubscriptdouble-integral𝜂superscriptnorm𝜃21superscript𝜂2𝑝𝑛2superscript𝜂superscriptnorm𝜃2𝛼1𝑝2subscript𝑓𝐺𝜂superscriptnorm𝑧𝜃21differential-d𝜃differential-d𝜂subscriptdouble-integral𝜂superscriptnorm𝜃21superscript𝜂2𝑝𝑛2superscript𝜂superscriptnorm𝜃2𝛼2𝑝2subscript𝑓𝐺𝜂superscriptnorm𝑧𝜃21differential-d𝜃differential-d𝜂\displaystyle=\frac{\iint_{\eta\|\theta\|^{2}\leq 1}\eta^{(2p+n)/2}\{\eta\|\theta\|^{2}\}^{\alpha+(1-p)/2}f_{G}(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\theta\mathrm{d}\eta}{\iint_{\eta\|\theta\|^{2}\leq 1}\eta^{(2p+n)/2}\{\eta\|\theta\|^{2}\}^{\alpha+(2-p)/2}f_{G}(\eta\{\|z-\theta\|^{2}+1\})\mathrm{d}\theta\mathrm{d}\eta}
=μ21η(p+n)/2{μ2}α+(1p)/2exp(η1/2zμ2/2η/2)dμdημ21η(p+n)/2{μ2}α+(2p)/2exp(η1/2zμ2/2η/2)dμdη.absentsubscriptdouble-integralsuperscriptnorm𝜇21superscript𝜂𝑝𝑛2superscriptsuperscriptnorm𝜇2𝛼1𝑝2superscriptnormsuperscript𝜂12𝑧𝜇22𝜂2differential-d𝜇differential-d𝜂subscriptdouble-integralsuperscriptnorm𝜇21superscript𝜂𝑝𝑛2superscriptsuperscriptnorm𝜇2𝛼2𝑝2superscriptnormsuperscript𝜂12𝑧𝜇22𝜂2differential-d𝜇differential-d𝜂\displaystyle=\frac{\iint_{\|\mu\|^{2}\leq 1}\eta^{(p+n)/2}\{\|\mu\|^{2}\}^{\alpha+(1-p)/2}\exp(-\|\eta^{1/2}z-\mu\|^{2}/2-\eta/2)\mathrm{d}\mu\mathrm{d}\eta}{\iint_{\|\mu\|^{2}\leq 1}\eta^{(p+n)/2}\{\|\mu\|^{2}\}^{\alpha+(2-p)/2}\exp(-\|\eta^{1/2}z-\mu\|^{2}/2-\eta/2)\mathrm{d}\mu\mathrm{d}\eta}.

Note μ2superscriptnorm𝜇2\|\mu\|^{2} may be regarded as a non-central chi-square random variable with p𝑝p degrees of freedom and ηz2𝜂superscriptnorm𝑧2\eta\|z\|^{2} non-centrality parameter. For

aj(ηz2)=1Γ(p/2+j)2p/2+j(ηz2/2)jj!exp(ηz2/2),subscript𝑎𝑗𝜂superscriptnorm𝑧21Γ𝑝2𝑗superscript2𝑝2𝑗superscript𝜂superscriptnorm𝑧22𝑗𝑗𝜂superscriptnorm𝑧22\displaystyle a_{j}(\eta\|z\|^{2})=\frac{1}{\Gamma(p/2+j)2^{p/2+j}}\frac{(\eta\|z\|^{2}/2)^{j}}{j!}\exp(-\eta\|z\|^{2}/2),

we have

J2(f,π,z)subscript𝐽2𝑓𝜋𝑧\displaystyle J_{2}(f,\pi,z)
=j=00η(p+n)/2aj(ηz2)exp(η/2)dη01rα1/2+jexp(r/2)drj=00η(p+n)/2aj(ηz2)exp(η/2)dη01rα+jexp(r/2)drabsentsuperscriptsubscript𝑗0superscriptsubscript0superscript𝜂𝑝𝑛2subscript𝑎𝑗𝜂superscriptnorm𝑧2𝜂2differential-d𝜂superscriptsubscript01superscript𝑟𝛼12𝑗𝑟2differential-d𝑟superscriptsubscript𝑗0superscriptsubscript0superscript𝜂𝑝𝑛2subscript𝑎𝑗𝜂superscriptnorm𝑧2𝜂2differential-d𝜂superscriptsubscript01superscript𝑟𝛼𝑗𝑟2differential-d𝑟\displaystyle=\frac{\sum_{j=0}^{\infty}\int_{0}^{\infty}\eta^{(p+n)/2}a_{j}(\eta\|z\|^{2})\exp(-\eta/2)\mathrm{d}\eta\int_{0}^{1}r^{\alpha-1/2+j}\exp(-r/2)\mathrm{d}r}{\sum_{j=0}^{\infty}\int_{0}^{\infty}\eta^{(p+n)/2}a_{j}(\eta\|z\|^{2})\exp(-\eta/2)\mathrm{d}\eta\int_{0}^{1}r^{\alpha+j}\exp(-r/2)\mathrm{d}r}
=j=0a~j(z2)E[Rj1/2]j=0a~j(z2)E[Rj],absentsuperscriptsubscript𝑗0subscript~𝑎𝑗superscriptnorm𝑧2𝐸delimited-[]superscript𝑅𝑗12superscriptsubscript𝑗0subscript~𝑎𝑗superscriptnorm𝑧2𝐸delimited-[]superscript𝑅𝑗\displaystyle=\frac{\sum_{j=0}^{\infty}\tilde{a}_{j}(\|z\|^{2})E[R^{j-1/2}]}{\sum_{j=0}^{\infty}\tilde{a}_{j}(\|z\|^{2})E[R^{j}]},

where the expected value is taken under the probability density given by

rαexp(r/2)I[0,1](r)01rαexp(r/2)drsuperscript𝑟𝛼𝑟2subscript𝐼01𝑟superscriptsubscript01superscript𝑟𝛼𝑟2differential-d𝑟\frac{r^{\alpha}\exp(-r/2)I_{[0,1]}(r)}{\int_{0}^{1}r^{\alpha}\exp(-r/2)\mathrm{d}r}

and

a~j(z2)subscript~𝑎𝑗superscriptnorm𝑧2\displaystyle\tilde{a}_{j}(\|z\|^{2}) =0η(p+n)/2aj(ηz2)exp(η/2)dηabsentsuperscriptsubscript0superscript𝜂𝑝𝑛2subscript𝑎𝑗𝜂superscriptnorm𝑧2𝜂2differential-d𝜂\displaystyle=\int_{0}^{\infty}\eta^{(p+n)/2}a_{j}(\eta\|z\|^{2})\exp(-\eta/2)\mathrm{d}\eta
=Γ((p+n)/2+j+1)2(p+n)/2+j+1Γ(p/2+j)2p/2+j(z2/2)jj!(z2+1)(p+n)/2+j+1.absentΓ𝑝𝑛2𝑗1superscript2𝑝𝑛2𝑗1Γ𝑝2𝑗superscript2𝑝2𝑗superscriptsuperscriptnorm𝑧22𝑗𝑗superscriptsuperscriptnorm𝑧21𝑝𝑛2𝑗1\displaystyle=\frac{\Gamma((p+n)/2+j+1)2^{(p+n)/2+j+1}}{\Gamma(p/2+j)2^{p/2+j}}\frac{(\|z\|^{2}/2)^{j}}{j!(\|z\|^{2}+1)^{(p+n)/2+j+1}}.

Since the correlation inequality gives

E[R1/2]E[R1/2]E[R]E[R3/2]E[R2],𝐸delimited-[]superscript𝑅12𝐸delimited-[]superscript𝑅12𝐸delimited-[]𝑅𝐸delimited-[]superscript𝑅32𝐸delimited-[]superscript𝑅2\displaystyle E[R^{-1/2}]\geq\frac{E[R^{1/2}]}{E[R]}\geq\frac{E[R^{3/2}]}{E[R^{2}]}\geq\dots,

we have

j=0a~j(z2)E[Rj1/2]j=0a~j(z2)E[Rj]E[R1/2]=01rα1/2exp(r/2)dr01rαexp(r/2)dr.superscriptsubscript𝑗0subscript~𝑎𝑗superscriptnorm𝑧2𝐸delimited-[]superscript𝑅𝑗12superscriptsubscript𝑗0subscript~𝑎𝑗superscriptnorm𝑧2𝐸delimited-[]superscript𝑅𝑗𝐸delimited-[]superscript𝑅12superscriptsubscript01superscript𝑟𝛼12𝑟2differential-d𝑟superscriptsubscript01superscript𝑟𝛼𝑟2differential-d𝑟\frac{\sum_{j=0}^{\infty}\tilde{a}_{j}(\|z\|^{2})E[R^{j-1/2}]}{\sum_{j=0}^{\infty}\tilde{a}_{j}(\|z\|^{2})E[R^{j}]}\leq E[R^{-1/2}]=\frac{\int_{0}^{1}r^{\alpha-1/2}\exp(-r/2)\mathrm{d}r}{\int_{0}^{1}r^{\alpha}\exp(-r/2)\mathrm{d}r}.

For 0r10𝑟10\leq r\leq 1, we have

1/2<exp(1/2)exp(r/2)1,1212𝑟21\displaystyle 1/2<\exp(-1/2)\leq\exp(-r/2)\leq 1,
E[R1/2]2α+1α+1/2,𝐸delimited-[]superscript𝑅122𝛼1𝛼12\displaystyle E[R^{-1/2}]\leq 2\frac{\alpha+1}{\alpha+1/2},

and hence

J2(f,π,z)2α+1α+1/2, for any zp.formulae-sequencesubscript𝐽2𝑓𝜋𝑧2𝛼1𝛼12 for any 𝑧superscript𝑝J_{2}(f,\pi,z)\leq 2\frac{\alpha+1}{\alpha+1/2},\text{ for any }z\in\mathbb{R}^{p}. (I.8)

Finally, by (I.4), (I.5), (I.6), (I.7) and (I.8), we have

J(f,π,z)2α+1α+1/2maxa+φ(a;α+1/2)mina+φ(a;α+1)maxλ[0,1]ν(λ)minλ[0,1]ν(λ).𝐽𝑓𝜋𝑧2𝛼1𝛼12subscript𝑎subscript𝜑𝑎𝛼12subscript𝑎subscript𝜑𝑎𝛼1subscript𝜆01𝜈𝜆subscript𝜆01𝜈𝜆\displaystyle J(f,\pi,z)\leq 2\frac{\alpha+1}{\alpha+1/2}\frac{\max_{a\in\mathbb{R}_{+}}\varphi(a;\alpha+1/2)}{\min_{a\in\mathbb{R}_{+}}\varphi(a;\alpha+1)}\frac{\max_{\lambda\in[0,1]}\nu(\lambda)}{\min_{\lambda\in[0,1]}\nu(\lambda)}.

Using Lemma I.1, we have the following result.

Lemma I.2.

Suppose Assumptions F.1F.3 on f𝑓f hold. Assume Assumptions A.2 on π𝜋\pi with 1/2<α012𝛼0-1/2<\alpha\leq 0. Let

k(λ)=λ1/2I[0,1](λ)+I(1,)(λ).𝑘𝜆superscript𝜆12subscript𝐼01𝜆subscript𝐼1𝜆\displaystyle k(\lambda)=\lambda^{1/2}I_{[0,1]}(\lambda)+I_{(1,\infty)}(\lambda).
  1. 1.

    Then

    η(2p+n)/2f(η{zθ2+1}){π¯(ηθ2)/k(ηθ2)}dθdηη(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη𝒥1(f,π),double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2𝑘𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂subscript𝒥1𝑓𝜋\begin{split}&\frac{\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\{\bar{\pi}(\eta\|\theta\|^{2})/k(\eta\|\theta\|^{2})\}\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}\\ &\leq\mathcal{J}_{1}(f,\pi),\end{split} (I.9)

    where

    𝒥1(f,π)=𝒥(f,π)+1subscript𝒥1𝑓𝜋𝒥𝑓𝜋1\mathcal{J}_{1}(f,\pi)=\mathcal{J}(f,\pi)+1 (I.10)

    and 𝒥(f,π)𝒥𝑓𝜋\mathcal{J}(f,\pi) is given by (I.3) of Lemma I.1.

  2. 2.

    We have

    η(2p+n)/2f(η{zθ2+1}){π¯i(ηθ2)/k(ηθ2)}dθdηη(2p+n)/2f(η{zθ2+1})π¯i(ηθ2)dθdη𝒥2(f,π),double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21subscript¯𝜋𝑖𝜂superscriptnorm𝜃2𝑘𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21subscript¯𝜋𝑖𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂subscript𝒥2𝑓𝜋\begin{split}&\frac{\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\{\bar{\pi}_{i}(\eta\|\theta\|^{2})/k(\eta\|\theta\|^{2})\}\mathrm{d}\theta\mathrm{d}\eta}{\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}_{i}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta}\\ &\leq\mathcal{J}_{2}(f,\pi),\end{split} (I.11)

    where

    𝒥2(f,π)=64𝒥(f,π)+1.subscript𝒥2𝑓𝜋64𝒥𝑓𝜋1\displaystyle\mathcal{J}_{2}(f,\pi)=64\mathcal{J}(f,\pi)+1.
Proof.

Let ={(θ,η):ηθ21}conditional-set𝜃𝜂𝜂superscriptnorm𝜃21\mathcal{R}=\{(\theta,\eta):\eta\|\theta\|^{2}\leq 1\}. The parameter space for (θ,η)𝜃𝜂(\theta,\eta) is decomposed as

p×+=C and C=.superscript𝑝subscriptsuperscript𝐶 and superscript𝐶\displaystyle\mathbb{R}^{p}\times\mathbb{R}_{+}=\mathcal{R}\cup\mathcal{R}^{C}\text{ and }\mathcal{R}\cap\mathcal{R}^{C}=\emptyset.

Then

η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)k(ηθ2)dθdη=(+C)η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)k(ηθ2)dθdη𝒥(f,π)η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη+Cη(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη{𝒥(f,π)+1}η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdη,double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2𝑘𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂subscriptdouble-integralsubscriptdouble-integralsuperscript𝐶superscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2𝑘𝜂superscriptnorm𝜃2d𝜃d𝜂𝒥𝑓𝜋subscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂subscriptdouble-integralsuperscript𝐶superscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂𝒥𝑓𝜋1double-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑧𝜃21¯𝜋𝜂superscriptdelimited-∥∥𝜃2differential-d𝜃differential-d𝜂\begin{split}&\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\frac{\bar{\pi}(\eta\|\theta\|^{2})}{k(\eta\|\theta\|^{2})}\mathrm{d}\theta\mathrm{d}\eta\\ &=\left(\iint_{\mathcal{R}}+\iint_{\mathcal{R}^{C}}\right)\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\frac{\bar{\pi}(\eta\|\theta\|^{2})}{k(\eta\|\theta\|^{2})}\mathrm{d}\theta\mathrm{d}\eta\\ &\leq\mathcal{J}(f,\pi)\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta\\ &\qquad+\iint_{\mathcal{R}^{C}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta\\ &\leq\left\{\mathcal{J}(f,\pi)+1\right\}\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta,\end{split} (I.12)

which completes the proof of Part 1.

For Part 2, note the following relationship;

η(2p+n)/2f(η{zθ2+1})π¯i(ηθ2)k(ηθ2)dθdηsubscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21subscript¯𝜋𝑖𝜂superscriptnorm𝜃2𝑘𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\frac{\bar{\pi}_{i}(\eta\|\theta\|^{2})}{k(\eta\|\theta\|^{2})}\mathrm{d}\theta\mathrm{d}\eta
η(2p+n)/2f(η{zθ2+1})π¯i(ηθ2)k(ηθ2)dθdηabsentsubscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21subscript¯𝜋𝑖𝜂superscriptnorm𝜃2𝑘𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\frac{\bar{\pi}_{i}(\eta\|\theta\|^{2})}{k(\eta\|\theta\|^{2})}\mathrm{d}\theta\mathrm{d}\eta
𝒥(f,π)η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdηabsent𝒥𝑓𝜋subscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq\mathcal{J}(f,\pi)\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
=𝒥(f,π)h12(1)η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)h12(1)dθdηabsent𝒥𝑓𝜋superscriptsubscript121subscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2superscriptsubscript121differential-d𝜃differential-d𝜂\displaystyle=\frac{\mathcal{J}(f,\pi)}{h_{1}^{2}(1)}\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})h_{1}^{2}(1)\mathrm{d}\theta\mathrm{d}\eta
𝒥(f,π)h12(1)η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)hi2(ηθ2)dθdηabsent𝒥𝑓𝜋superscriptsubscript121subscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2superscriptsubscript𝑖2𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq\frac{\mathcal{J}(f,\pi)}{h_{1}^{2}(1)}\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})h_{i}^{2}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
64𝒥(f,π)η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)hi2(ηθ2)dθdη.absent64𝒥𝑓𝜋subscriptdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2superscriptsubscript𝑖2𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\leq 64\mathcal{J}(f,\pi)\iint_{\mathcal{R}}\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})h_{i}^{2}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta.

where the first inequality follows from the fact hi21superscriptsubscript𝑖21h_{i}^{2}\leq 1, the second inequality follows from Lemma I.1, the third inequality follows from Part 1 of Lemma E.2. The last inequality follows from Part 4 of Lemma E.2. Then, as in (I.12), the inequality (I.11) can be established. ∎

Lemma I.3.

Under Assumptions F.1F.3 on f𝑓f and Assumptions A.1, A.2 A.3 on π𝜋\pi,

zTη(2p+n)/21F(η{zθ2+1})θπ¯(ηθ2)dθdηsuperscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21subscript𝜃¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
=η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdηabsentdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
(p+n)η(2p+n)/21F(η{zθ2+1})π¯(ηθ2)dθdη.𝑝𝑛double-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\qquad-(p+n)\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta.
Proof.

First, note the following relationship;

zTη(2p+n)/21F(η{zθ2+1})θπ¯(ηθ2)dθdηsuperscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21subscript𝜃¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
=zTη(2p+n)/21F(η{zθ2+1})2θηπ¯(ηθ2)dθdηabsentsuperscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃212𝜃𝜂superscript¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})2\theta\eta\bar{\pi}^{\prime}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
=2zTη(p+n)/2F(η1/2zμ2+η)η1/2μπ¯(μ2)dμdηabsent2superscript𝑧Tdouble-integralsuperscript𝜂𝑝𝑛2𝐹superscriptnormsuperscript𝜂12𝑧𝜇2𝜂superscript𝜂12𝜇superscript¯𝜋superscriptnorm𝜇2differential-d𝜇differential-d𝜂\displaystyle=2z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(p+n)/2}F(\|\eta^{1/2}z-\mu\|^{2}+\eta)\eta^{-1/2}\mu\bar{\pi}^{\prime}(\|\mu\|^{2})\mathrm{d}\mu\mathrm{d}\eta
=2η(p+n)/2F(μ2+η)η1/2zT(μ+η1/2z)π¯(μ+η1/2z2)dμdηabsent2double-integralsuperscript𝜂𝑝𝑛2𝐹superscriptnorm𝜇2𝜂superscript𝜂12superscript𝑧T𝜇superscript𝜂12𝑧superscript¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜇differential-d𝜂\displaystyle=2\iint\eta^{(p+n)/2}F(\|\mu\|^{2}+\eta)\eta^{-1/2}z^{\mathrm{\scriptscriptstyle T}}(\mu+\eta^{1/2}z)\bar{\pi}^{\prime}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\mu\mathrm{d}\eta
=2η(p+n)/2F(μ2+η)ηπ¯(μ+η1/2z2)dμdη.absent2double-integralsuperscript𝜂𝑝𝑛2𝐹superscriptnorm𝜇2𝜂𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜇differential-d𝜂\displaystyle=2\iint\eta^{(p+n)/2}F(\|\mu\|^{2}+\eta)\frac{\partial}{\partial\eta}\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\mu\mathrm{d}\eta.

By an integration by parts, the integral with respect to η𝜂\eta in the above is

0η(p+n)/2F(μ2+η)ηπ¯(μ+η1/2z2)dηsuperscriptsubscript0superscript𝜂𝑝𝑛2𝐹superscriptnorm𝜇2𝜂𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜂\displaystyle\int_{0}^{\infty}\eta^{(p+n)/2}F(\|\mu\|^{2}+\eta)\frac{\partial}{\partial\eta}\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\eta
=[η(p+n)/2F(μ2+η)π¯(μ+η1/2z2)]0absentsuperscriptsubscriptdelimited-[]superscript𝜂𝑝𝑛2𝐹superscriptnorm𝜇2𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧20\displaystyle=\left[\eta^{(p+n)/2}F(\|\mu\|^{2}+\eta)\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\right]_{0}^{\infty}
+120η(p+n)/2f(μ2+η)π¯(μ+η1/2z2)dη12superscriptsubscript0superscript𝜂𝑝𝑛2𝑓superscriptnorm𝜇2𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜂\displaystyle\quad+\frac{1}{2}\int_{0}^{\infty}\eta^{(p+n)/2}f(\|\mu\|^{2}+\eta)\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\eta
p+n20η(p+n)/21F(μ2+η)π¯(μ+η1/2z2)dμdη,𝑝𝑛2superscriptsubscript0superscript𝜂𝑝𝑛21𝐹superscriptnorm𝜇2𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜇differential-d𝜂\displaystyle\quad-\frac{p+n}{2}\int_{0}^{\infty}\eta^{(p+n)/2-1}F(\|\mu\|^{2}+\eta)\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\mu\mathrm{d}\eta,

where the first term becomes zero for any fixed μ𝜇\mu under Assumptions. Then

zTη(2p+n)/21F(η{zθ2+1})θπ¯(ηθ2)dθdηsuperscript𝑧Tdouble-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21subscript𝜃¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle z^{\mathrm{\scriptscriptstyle T}}\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\nabla_{\theta}\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
=η(p+n)/2f(μ2+η)π¯(μ+η1/2z2)dηabsentdouble-integralsuperscript𝜂𝑝𝑛2𝑓superscriptnorm𝜇2𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜂\displaystyle=\iint\eta^{(p+n)/2}f(\|\mu\|^{2}+\eta)\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\eta
(p+n)η(p+n)/21F(μ2+η)π¯(μ+η1/2z2)dμdη𝑝𝑛double-integralsuperscript𝜂𝑝𝑛21𝐹superscriptnorm𝜇2𝜂¯𝜋superscriptnorm𝜇superscript𝜂12𝑧2differential-d𝜇differential-d𝜂\displaystyle\qquad-(p+n)\iint\eta^{(p+n)/2-1}F(\|\mu\|^{2}+\eta)\bar{\pi}(\|\mu+\eta^{1/2}z\|^{2})\mathrm{d}\mu\mathrm{d}\eta
=η(2p+n)/2f(η{zθ2+1})π¯(ηθ2)dθdηabsentdouble-integralsuperscript𝜂2𝑝𝑛2𝑓𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle=\iint\eta^{(2p+n)/2}f(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta
(p+n)η(2p+n)/21F(η{zθ2+1})π¯(ηθ2)dθdη,𝑝𝑛double-integralsuperscript𝜂2𝑝𝑛21𝐹𝜂superscriptnorm𝑧𝜃21¯𝜋𝜂superscriptnorm𝜃2differential-d𝜃differential-d𝜂\displaystyle\qquad-(p+n)\iint\eta^{(2p+n)/2-1}F(\eta\{\|z-\theta\|^{2}+1\})\bar{\pi}(\eta\|\theta\|^{2})\mathrm{d}\theta\mathrm{d}\eta,

which completes the proof. ∎

Appendix J Proof of Corollary 4.2, Part 3

Let ϕα(w)=wψα(w)subscriptitalic-ϕ𝛼𝑤𝑤subscript𝜓𝛼𝑤\phi_{\alpha}(w)=w\psi_{\alpha}(w) where, as in (4.11),

ψα(w)=01tp/2α1(1t)α(1+wt)(p+n)/21dt01tp/2α2(1t)α(1+wt)(p+n)/21dt,subscript𝜓𝛼𝑤superscriptsubscript01superscript𝑡𝑝2𝛼1superscript1𝑡𝛼superscript1𝑤𝑡𝑝𝑛21differential-d𝑡superscriptsubscript01superscript𝑡𝑝2𝛼2superscript1𝑡𝛼superscript1𝑤𝑡𝑝𝑛21differential-d𝑡\psi_{\alpha}(w)=\frac{\int_{0}^{1}t^{p/2-\alpha-1}(1-t)^{\alpha}(1+wt)^{-(p+n)/2-1}\mathrm{d}t}{\int_{0}^{1}t^{p/2-\alpha-2}(1-t)^{\alpha}(1+wt)^{-(p+n)/2-1}\mathrm{d}t},

for α(1,0)𝛼10\alpha\in(-1,0). Maruyama and Strawderman (2009) showed that

  1. 1.

    ϕα(w)subscriptitalic-ϕ𝛼𝑤\phi_{\alpha}(w) is not monotonic,

  2. 2.

    0ϕα(w)ϕ(α)0subscriptitalic-ϕ𝛼𝑤subscriptitalic-ϕ𝛼0\leq\phi_{\alpha}(w)\leq\phi_{\star}(\alpha)

    ϕ(α)=p/2α1n/2+α+1+α(p/2+n/2),subscriptitalic-ϕ𝛼𝑝2𝛼1𝑛2𝛼1𝛼𝑝2𝑛2\phi_{\star}(\alpha)=\frac{p/2-\alpha-1}{n/2+\alpha+1+\alpha(p/2+n/2)},
  3. 3.

    wϕα(w)/ϕα(w)c(α)𝑤subscriptsuperscriptitalic-ϕ𝛼𝑤subscriptitalic-ϕ𝛼𝑤𝑐𝛼w\phi^{\prime}_{\alpha}(w)/\phi_{\alpha}(w)\geq-c(\alpha)

    c(α)=(p/2α)αα+1.𝑐𝛼𝑝2𝛼𝛼𝛼1c(\alpha)=-\frac{(p/2-\alpha)\alpha}{\alpha+1}.

in Part (iii) of Corollary 3.1, Part (iv) of Corollary 3.1 and Lemma 3.4, respectively. Kubokawa (2009) proposed a sufficient condition of δϕ={1ϕ(W)/W}Xsubscript𝛿italic-ϕ1italic-ϕ𝑊𝑊𝑋\delta_{\phi}=\{1-\phi(W)/W\}X to be minimax as follows;

wϕ(w)/ϕ(w)c, and 0ϕ2p22cn+2+2c, for some c>0,formulae-sequenceformulae-sequence𝑤superscriptitalic-ϕ𝑤italic-ϕ𝑤𝑐 and 0italic-ϕ2𝑝22𝑐𝑛22𝑐 for some 𝑐0w\phi^{\prime}(w)/\phi(w)\geq-c,\text{ and }0\leq\phi\leq 2\frac{p-2-2c}{n+2+2c},\text{ for some }c>0,

where the upper bound is larger than the upper bound which Maruyama and Strawderman (2009) and Wells and Zhou (2008) applied. Note that ϕ(α)subscriptitalic-ϕ𝛼\phi_{\star}(\alpha) is increasing in α(1/2,0)𝛼120\alpha\in(-1/2,0) and that c(α)𝑐𝛼c(\alpha) is decreasing in α(1/2,0)𝛼120\alpha\in(-1/2,0). Then the inequalities

p/2α1n/2+α+1+α(p/2+n/2)2p22c(α)n+2+2c(α)=2(p2)(α+1)+2(p/2α)α(n+2)(α+1)2(p/2α)α𝑝2𝛼1𝑛2𝛼1𝛼𝑝2𝑛22𝑝22𝑐𝛼𝑛22𝑐𝛼2𝑝2𝛼12𝑝2𝛼𝛼𝑛2𝛼12𝑝2𝛼𝛼\begin{split}\frac{p/2-\alpha-1}{n/2+\alpha+1+\alpha(p/2+n/2)}&\leq 2\frac{p-2-2c(\alpha)}{n+2+2c(\alpha)}\\ &=2\frac{(p-2)(\alpha+1)+2(p/2-\alpha)\alpha}{(n+2)(\alpha+1)-2(p/2-\alpha)\alpha}\end{split}

as well as 1<α<01𝛼0-1<\alpha<0 are a sufficient condition for minimaxity of δψαsubscript𝛿subscript𝜓𝛼\delta_{\psi_{\alpha}}. Let

f(α)𝑓𝛼\displaystyle f(\alpha) =2{(p2)(α+1)+2(p/2α)α}{n/2+α+1+α(p/2+n/2)}absent2𝑝2𝛼12𝑝2𝛼𝛼𝑛2𝛼1𝛼𝑝2𝑛2\displaystyle=2\left\{(p-2)(\alpha+1)+2(p/2-\alpha)\alpha\right\}\left\{n/2+\alpha+1+\alpha(p/2+n/2)\right\}
{(n+2)(α+1)2(p/2α)α}(p/2α1)𝑛2𝛼12𝑝2𝛼𝛼𝑝2𝛼1\displaystyle\quad-\left\{(n+2)(\alpha+1)-2(p/2-\alpha)\alpha\right\}\left(p/2-\alpha-1\right)
=(p2)(n+2)2+(n+2)(5p8)+3p(p2)2αabsent𝑝2𝑛22𝑛25𝑝83𝑝𝑝22𝛼\displaystyle=\frac{(p-2)(n+2)}{2}+\frac{(n+2)(5p-8)+3p(p-2)}{2}\alpha
+{2(p1)2+(2p3)(n+2)}α22(p+n+1)α3.2superscript𝑝122𝑝3𝑛2superscript𝛼22𝑝𝑛1superscript𝛼3\displaystyle\quad+\left\{2(p-1)^{2}+(2p-3)(n+2)\right\}\alpha^{2}-2(p+n+1)\alpha^{3}.

For α(1/2,0)𝛼120\alpha\in(-1/2,0),

f(α)(p2)(n+2)2+5(n+2)(p2)+2(n+2)+3p(p2)2α𝑓𝛼𝑝2𝑛225𝑛2𝑝22𝑛23𝑝𝑝22𝛼f(\alpha)\geq\frac{(p-2)(n+2)}{2}+\frac{5(n+2)(p-2)+2(n+2)+3p(p-2)}{2}\alpha

which is nonnegative when

(5+2p2+3pn+2)1α<0.superscript52𝑝23𝑝𝑛21𝛼0-\left(5+\frac{2}{p-2}+\frac{3p}{n+2}\right)^{-1}\leq\alpha<0.

Hence Part 3 follows.

Appendix K Proof of Corollary 4.3

Let

π¯(ηθ2)=b1(2π)p/2ξp/2exp(ηθ22ξ)gπ(ξ)dξ¯𝜋𝜂superscriptnorm𝜃2superscriptsubscript𝑏1superscript2𝜋𝑝2superscript𝜉𝑝2𝜂superscriptnorm𝜃22𝜉subscript𝑔𝜋𝜉differential-d𝜉\bar{\pi}(\eta\|\theta\|^{2})=\int_{b}^{\infty}\frac{1}{(2\pi)^{p/2}\xi^{p/2}}\exp\left(-\frac{\eta\|\theta\|^{2}}{2\xi}\right)g_{\pi}(\xi)\mathrm{d}\xi

where

gπ(ξ)=(ξb)α(ξ+1)β.subscript𝑔𝜋𝜉superscript𝜉𝑏𝛼superscript𝜉1𝛽\displaystyle g_{\pi}(\xi)=(\xi-b)^{\alpha}(\xi+1)^{\beta}.

Eventually set 1<αn/21𝛼𝑛2-1<\alpha\leq n/2, β=n/2𝛽𝑛2\beta=-n/2 and b0𝑏0b\geq 0.

Note the underlying density is Gaussian and let

fG(t)=1(2π)(p+n)/2exp(t/2).subscript𝑓𝐺𝑡1superscript2𝜋𝑝𝑛2𝑡2\displaystyle f_{G}(t)=\frac{1}{(2\pi)^{(p+n)/2}}\exp(-t/2).

Note

zθ2+θ2ξ=ξ+1ξθξξ+1z2+z2ξ+1.superscriptnorm𝑧𝜃2superscriptnorm𝜃2𝜉𝜉1𝜉superscriptnorm𝜃𝜉𝜉1𝑧2superscriptnorm𝑧2𝜉1\displaystyle\|z-\theta\|^{2}+\frac{\|\theta\|^{2}}{\xi}=\frac{\xi+1}{\xi}\left\|\theta-\frac{\xi}{\xi+1}z\right\|^{2}+\frac{\|z\|^{2}}{\xi+1}.

Then we have

M1(z;π)subscript𝑀1𝑧𝜋\displaystyle M_{1}(z;\pi)
=1(2π)(p+n)/2b0η(p+n)/2exp(η2)gπ(ξ)absent1superscript2𝜋𝑝𝑛2superscriptsubscript𝑏superscriptsubscript0superscript𝜂𝑝𝑛2𝜂2subscript𝑔𝜋𝜉\displaystyle=\frac{1}{(2\pi)^{(p+n)/2}}\int_{b}^{\infty}\int_{0}^{\infty}\eta^{(p+n)/2}\exp\left(-\frac{\eta}{2}\right)g_{\pi}(\xi)
×{pηp/2(2π)p/2ξp/2exp(ηzθ22ηθ22ξ)dθ}dηdξabsentsubscriptsuperscript𝑝superscript𝜂𝑝2superscript2𝜋𝑝2superscript𝜉𝑝2𝜂superscriptnorm𝑧𝜃22𝜂superscriptnorm𝜃22𝜉differential-d𝜃d𝜂d𝜉\displaystyle\quad\times\left\{\int_{\mathbb{R}^{p}}\frac{\eta^{p/2}}{(2\pi)^{p/2}\xi^{p/2}}\exp\left(-\eta\frac{\|z-\theta\|^{2}}{2}-\frac{\eta\|\theta\|^{2}}{2\xi}\right)\mathrm{d}\theta\right\}\mathrm{d}\eta\mathrm{d}\xi
=1(2π)(p+n)/2b0η(p+n)/2exp(η2)gπ(ξ)absent1superscript2𝜋𝑝𝑛2superscriptsubscript𝑏superscriptsubscript0superscript𝜂𝑝𝑛2𝜂2subscript𝑔𝜋𝜉\displaystyle=\frac{1}{(2\pi)^{(p+n)/2}}\int_{b}^{\infty}\int_{0}^{\infty}\eta^{(p+n)/2}\exp\left(-\frac{\eta}{2}\right)g_{\pi}(\xi)
×1(ξ+1)p/2exp(ηz22(ξ+1))dηdξabsent1superscript𝜉1𝑝2𝜂superscriptnorm𝑧22𝜉1d𝜂d𝜉\displaystyle\quad\times\frac{1}{(\xi+1)^{p/2}}\exp\left(-\frac{\eta\|z\|^{2}}{2(\xi+1)}\right)\mathrm{d}\eta\mathrm{d}\xi
=cb(1+z2ξ+1)(p+n)/21gπ(ξ)(ξ+1)p/2dξabsent𝑐superscriptsubscript𝑏superscript1superscriptnorm𝑧2𝜉1𝑝𝑛21subscript𝑔𝜋𝜉superscript𝜉1𝑝2differential-d𝜉\displaystyle=c\int_{b}^{\infty}\left(1+\frac{\|z\|^{2}}{\xi+1}\right)^{-(p+n)/2-1}\frac{g_{\pi}(\xi)}{(\xi+1)^{p/2}}\mathrm{d}\xi
=cb(ξb)α(1+ξ)β+n/2+1(1+ξ+z2)(p+n)/2+1dξabsent𝑐superscriptsubscript𝑏superscript𝜉𝑏𝛼superscript1𝜉𝛽𝑛21superscript1𝜉superscriptnorm𝑧2𝑝𝑛21differential-d𝜉\displaystyle=c\int_{b}^{\infty}\frac{(\xi-b)^{\alpha}(1+\xi)^{\beta+n/2+1}}{(1+\xi+\|z\|^{2})^{(p+n)/2+1}}\mathrm{d}\xi
=c0ξα(1+b+ξ)β+n/2+1(1+b+z2+ξ)(p+n)/2+1dξabsent𝑐superscriptsubscript0superscript𝜉𝛼superscript1𝑏𝜉𝛽𝑛21superscript1𝑏superscriptnorm𝑧2𝜉𝑝𝑛21differential-d𝜉\displaystyle=c\int_{0}^{\infty}\frac{\xi^{\alpha}(1+b+\xi)^{\beta+n/2+1}}{(1+b+\|z\|^{2}+\xi)^{(p+n)/2+1}}\mathrm{d}\xi

where

c=Γ((p+n)/2+1)2(p+n)/2+1(2π)(p+n)/2.𝑐Γ𝑝𝑛21superscript2𝑝𝑛21superscript2𝜋𝑝𝑛2\displaystyle c=\frac{\Gamma((p+n)/2+1)2^{(p+n)/2+1}}{(2\pi)^{(p+n)/2}}.

Also we have

zTM2(z;π)superscript𝑧Tsubscript𝑀2𝑧𝜋\displaystyle z^{\mathrm{\scriptscriptstyle T}}M_{2}(z;\pi)
=1(2π)(p+n)/2b0η(p+n)/2exp(η2)gπ(ξ)absent1superscript2𝜋𝑝𝑛2superscriptsubscript𝑏superscriptsubscript0superscript𝜂𝑝𝑛2𝜂2subscript𝑔𝜋𝜉\displaystyle=\frac{1}{(2\pi)^{(p+n)/2}}\int_{b}^{\infty}\int_{0}^{\infty}\eta^{(p+n)/2}\exp\left(-\frac{\eta}{2}\right)g_{\pi}(\xi)
×{pzTθηp/2(2π)p/2ξp/2exp(ηzθ22ηθ22ξ)dθ}dηdξabsentsubscriptsuperscript𝑝superscript𝑧T𝜃superscript𝜂𝑝2superscript2𝜋𝑝2superscript𝜉𝑝2𝜂superscriptnorm𝑧𝜃22𝜂superscriptnorm𝜃22𝜉differential-d𝜃d𝜂d𝜉\displaystyle\quad\times\left\{\int_{\mathbb{R}^{p}}\frac{z^{\mathrm{\scriptscriptstyle T}}\theta\eta^{p/2}}{(2\pi)^{p/2}\xi^{p/2}}\exp\left(-\eta\frac{\|z-\theta\|^{2}}{2}-\frac{\eta\|\theta\|^{2}}{2\xi}\right)\mathrm{d}\theta\right\}\mathrm{d}\eta\mathrm{d}\xi
=z2(2π)(p+n)/2b0η(p+n)/2exp(η2)gπ(ξ)absentsuperscriptnorm𝑧2superscript2𝜋𝑝𝑛2superscriptsubscript𝑏superscriptsubscript0superscript𝜂𝑝𝑛2𝜂2subscript𝑔𝜋𝜉\displaystyle=\frac{\|z\|^{2}}{(2\pi)^{(p+n)/2}}\int_{b}^{\infty}\int_{0}^{\infty}\eta^{(p+n)/2}\exp\left(-\frac{\eta}{2}\right)g_{\pi}(\xi)
×1(ξ+1)p/2ξξ+1exp(ηz22(ξ+1))dηdξabsent1superscript𝜉1𝑝2𝜉𝜉1𝜂superscriptnorm𝑧22𝜉1d𝜂d𝜉\displaystyle\quad\times\frac{1}{(\xi+1)^{p/2}}\frac{\xi}{\xi+1}\exp\left(-\frac{\eta\|z\|^{2}}{2(\xi+1)}\right)\mathrm{d}\eta\mathrm{d}\xi
=z2M1(z;π)z2c0ξα(1+b+ξ)β+n/2(1+b+z2+ξ)(p+n)/2+1dξ.absentsuperscriptnorm𝑧2subscript𝑀1𝑧𝜋superscriptnorm𝑧2𝑐superscriptsubscript0superscript𝜉𝛼superscript1𝑏𝜉𝛽𝑛2superscript1𝑏superscriptnorm𝑧2𝜉𝑝𝑛21differential-d𝜉\displaystyle=\|z\|^{2}M_{1}(z;\pi)-\|z\|^{2}c\int_{0}^{\infty}\frac{\xi^{\alpha}(1+b+\xi)^{\beta+n/2}}{(1+b+\|z\|^{2}+\xi)^{(p+n)/2+1}}\mathrm{d}\xi.

Recall

ψπ(z)=zTzM1(z,π)zTM2(z,π)z2M1(z,π).subscript𝜓𝜋𝑧superscript𝑧T𝑧subscript𝑀1𝑧𝜋superscript𝑧Tsubscript𝑀2𝑧𝜋superscriptnorm𝑧2subscript𝑀1𝑧𝜋\displaystyle\psi_{\pi}(z)=\frac{z^{\mathrm{\scriptscriptstyle T}}zM_{1}(z,\pi)-z^{\mathrm{\scriptscriptstyle T}}M_{2}(z,\pi)}{\|z\|^{2}M_{1}(z,\pi)}.

Under the choice β=n/2𝛽𝑛2\beta=-n/2, we have

ψπ(z)subscript𝜓𝜋𝑧\displaystyle\psi_{\pi}(z) =0ξα(1+b+z2+ξ)(p+n)/21dξ0ξα(1+b+ξ)(1+b+z2+ξ)(p+n)/21dξabsentsuperscriptsubscript0superscript𝜉𝛼superscript1𝑏superscriptnorm𝑧2𝜉𝑝𝑛21differential-d𝜉superscriptsubscript0superscript𝜉𝛼1𝑏𝜉superscript1𝑏superscriptnorm𝑧2𝜉𝑝𝑛21differential-d𝜉\displaystyle=\frac{\int_{0}^{\infty}\xi^{\alpha}(1+b+\|z\|^{2}+\xi)^{-(p+n)/2-1}\mathrm{d}\xi}{\int_{0}^{\infty}\xi^{\alpha}(1+b+\xi)(1+b+\|z\|^{2}+\xi)^{-(p+n)/2-1}\mathrm{d}\xi}
=(1+c+0ξα+1(1+b+z2+ξ)(p+n)/21dξ0ξα(1+b+z2+ξ)(p+n)/21dξ)1absentsuperscript1𝑐superscriptsubscript0superscript𝜉𝛼1superscript1𝑏superscriptnorm𝑧2𝜉𝑝𝑛21differential-d𝜉superscriptsubscript0superscript𝜉𝛼superscript1𝑏superscriptnorm𝑧2𝜉𝑝𝑛21differential-d𝜉1\displaystyle=\left(1+c+\frac{\int_{0}^{\infty}\xi^{\alpha+1}(1+b+\|z\|^{2}+\xi)^{-(p+n)/2-1}\mathrm{d}\xi}{\int_{0}^{\infty}\xi^{\alpha}(1+b+\|z\|^{2}+\xi)^{-(p+n)/2-1}\mathrm{d}\xi}\right)^{-1}
=(1+b+(1+b+z2)B(α+2,(p+n)/2α1)B(α+1,(p+n)/2α))1absentsuperscript1𝑏1𝑏superscriptnorm𝑧2𝐵𝛼2𝑝𝑛2𝛼1𝐵𝛼1𝑝𝑛2𝛼1\displaystyle=\left(1+b+(1+b+\|z\|^{2})\frac{B(\alpha+2,(p+n)/2-\alpha-1)}{B(\alpha+1,(p+n)/2-\alpha)}\right)^{-1}
=(1+b+(1+b+z2)α+1(p+n)/2α1)1.absentsuperscript1𝑏1𝑏superscriptnorm𝑧2𝛼1𝑝𝑛2𝛼11\displaystyle=\left(1+b+(1+b+\|z\|^{2})\frac{\alpha+1}{(p+n)/2-\alpha-1}\right)^{-1}.

Let a={(p+n)/2α1}/(α+1)𝑎𝑝𝑛2𝛼1𝛼1a=\{(p+n)/2-\alpha-1\}/(\alpha+1). Then the Bayes equivariant estimator is

(1aX2/S+(a+1)(b+1))X.1𝑎superscriptnorm𝑋2𝑆𝑎1𝑏1𝑋\displaystyle\left(1-\frac{a}{\|X\|^{2}/S+(a+1)(b+1)}\right)X. (K.1)

When α+β<1𝛼𝛽1\alpha+\beta<-1 or equivalently α<n/21𝛼𝑛21\alpha<n/2-1 as well as α>1𝛼1\alpha>-1, this is a proper Bayes equivariant estimator. When 1α+β01𝛼𝛽0-1\leq\alpha+\beta\leq 0 or equivalently n/21αn/2𝑛21𝛼𝑛2n/2-1\leq\alpha\leq n/2 as well as α>1/2𝛼12\alpha>-1/2, this is an admissible generalized Bayes equivariant estimator. Hence when a(p2)/(n+2)𝑎𝑝2𝑛2a\geq(p-2)/(n+2) and b0𝑏0b\geq 0, the estimator (K.1) is admissible within the class of equivariant estimators.

Appendix L Proof of satisfaction of A.1A.3 by (4.4)

Lemma L.1.

Let π(λ)=cpλp/21π¯(λ)𝜋𝜆subscript𝑐𝑝superscript𝜆𝑝21¯𝜋𝜆\pi(\lambda)=c_{p}\lambda^{p/2-1}\bar{\pi}(\lambda) where

π¯(λ)=b1(2πξ)p/2exp(λ2ξ)(ξb)α(1+ξ)βdξ.¯𝜋𝜆superscriptsubscript𝑏1superscript2𝜋𝜉𝑝2𝜆2𝜉superscript𝜉𝑏𝛼superscript1𝜉𝛽differential-d𝜉\bar{\pi}(\lambda)=\int_{b}^{\infty}\frac{1}{(2\pi\xi)^{p/2}}\exp\left(-\frac{\lambda}{2\xi}\right)(\xi-b)^{\alpha}(1+\xi)^{\beta}\mathrm{d}\xi. (L.1)

Then Assumptions A.1A.3 are satisfied when {b>0,1α+β0, and α>1}formulae-sequenceformulae-sequence𝑏01𝛼𝛽0 and 𝛼1\{b>0,\ -1\leq\alpha+\beta\leq 0,\text{ and }\alpha>-1\} or {b=0,1α+β0, and α>1/2}formulae-sequenceformulae-sequence𝑏01𝛼𝛽0 and 𝛼12\{b=0,\ -1\leq\alpha+\beta\leq 0,\text{ and }\alpha>-1/2\}.

Note the integrability of (L.1) under (b,b+ϵ)𝑏𝑏italic-ϵ(b,b+\epsilon) follows b=0𝑏0b=0 or b>0𝑏0b>0 as well as α>1𝛼1\alpha>-1. Also the integrability of (L.1) under (b+ϵ,)𝑏italic-ϵ(b+\epsilon,\infty) follows when α+βp/2<1𝛼𝛽𝑝21\alpha+\beta-p/2<-1. Further note Note also if α+β<1𝛼𝛽1\alpha+\beta<-1 as well as α>1𝛼1\alpha>-1, b(ξb)α(1+ξ)βdξ<superscriptsubscript𝑏superscript𝜉𝑏𝛼superscript1𝜉𝛽differential-d𝜉\int_{b}^{\infty}(\xi-b)^{\alpha}(1+\xi)^{\beta}\mathrm{d}\xi<\infty and hence the prior on λ𝜆\lambda is proper.

Proof.

Clearly π(λ)𝜋𝜆\pi(\lambda) is differentiable as

π¯(λ)=12b(2π)p/2ξp/2+1exp(λ2ξ)(ξb)α(1+ξ)βdξ.superscript¯𝜋𝜆12superscriptsubscript𝑏superscript2𝜋𝑝2superscript𝜉𝑝21𝜆2𝜉superscript𝜉𝑏𝛼superscript1𝜉𝛽differential-d𝜉\bar{\pi}^{\prime}(\lambda)=-\frac{1}{2}\int_{b}^{\infty}\frac{(2\pi)^{-p/2}}{\xi^{p/2+1}}\exp\left(-\frac{\lambda}{2\xi}\right)(\xi-b)^{\alpha}(1+\xi)^{\beta}\mathrm{d}\xi. (L.2)

and hence Assumption A.1 is satisfied.

[Assumption A.2 with α>1/2𝛼12\alpha>-1/2 and b=0𝑏0b=0] When b=0𝑏0b=0, by Tauberian theorem, we have, in (L.1) and (L.2),

limλ0(2π)p/2π¯(λ)(λ/2)(p/2α1)Γ(p/2α1)=1limλ02(2π)p/2π¯(λ)(λ/2)(p/2α)Γ(p/2α)=1,subscript𝜆0superscript2𝜋𝑝2¯𝜋𝜆superscript𝜆2𝑝2𝛼1Γ𝑝2𝛼11subscript𝜆02superscript2𝜋𝑝2superscript¯𝜋𝜆superscript𝜆2𝑝2𝛼Γ𝑝2𝛼1\begin{split}\lim_{\lambda\to 0}\frac{(2\pi)^{p/2}\bar{\pi}(\lambda)}{(\lambda/2)^{-(p/2-\alpha-1)}\Gamma(p/2-\alpha-1)}=1\\ \lim_{\lambda\to 0}\frac{-2(2\pi)^{p/2}\bar{\pi}^{\prime}(\lambda)}{(\lambda/2)^{-(p/2-\alpha)}\Gamma(p/2-\alpha)}=1,\end{split} (L.3)

which implies that

limλ0λπ¯(λ)π¯(λ)=p2+α+1.subscript𝜆0𝜆superscript¯𝜋𝜆¯𝜋𝜆𝑝2𝛼1\lim_{\lambda\to 0}\lambda\frac{\bar{\pi}^{\prime}(\lambda)}{\bar{\pi}(\lambda)}=-\frac{p}{2}+\alpha+1. (L.4)

Recall π(λ)=cpλp/21π¯(λ)𝜋𝜆subscript𝑐𝑝superscript𝜆𝑝21¯𝜋𝜆\pi(\lambda)=c_{p}\lambda^{p/2-1}\bar{\pi}(\lambda) and let ν(λ)=λαπ(λ)=cpλp/21απ¯(λ)𝜈𝜆superscript𝜆𝛼𝜋𝜆subscript𝑐𝑝superscript𝜆𝑝21𝛼¯𝜋𝜆\nu(\lambda)=\lambda^{-\alpha}\pi(\lambda)=c_{p}\lambda^{p/2-1-\alpha}\bar{\pi}(\lambda). Then we have

ν(0)=cp2p/2α1Γ(p/2α1)(2π)p/2𝜈0subscript𝑐𝑝superscript2𝑝2𝛼1Γ𝑝2𝛼1superscript2𝜋𝑝2\displaystyle\nu(0)=c_{p}2^{p/2-\alpha-1}\Gamma(p/2-\alpha-1)(2\pi)^{-p/2}

by (L.3) and

limλ0λν(λ)ν(λ)=limλ0λν(λ)=0subscript𝜆0𝜆superscript𝜈𝜆𝜈𝜆subscript𝜆0𝜆superscript𝜈𝜆0\displaystyle\lim_{\lambda\to 0}\lambda\frac{\nu^{\prime}(\lambda)}{\nu(\lambda)}=\lim_{\lambda\to 0}\lambda\nu^{\prime}(\lambda)=0

by (L.4).

[Assumption A.2 with α>1𝛼1\alpha>-1 and b>0𝑏0b>0] When α>1𝛼1\alpha>-1 and b>0𝑏0b>0, it follows that 0<π¯(0)<0¯𝜋00<\bar{\pi}(0)<\infty and 0<|π¯(0)|<0superscript¯𝜋00<|\bar{\pi}^{\prime}(0)|<\infty. For this case take ν(λ)=cpπ¯(λ)𝜈𝜆subscript𝑐𝑝¯𝜋𝜆\nu(\lambda)=c_{p}\bar{\pi}(\lambda). Then π(λ)=λp/21ν(λ)𝜋𝜆superscript𝜆𝑝21𝜈𝜆\pi(\lambda)=\lambda^{p/2-1}\nu(\lambda), where p/21>1/2𝑝2112p/2-1>-1/2 and ν(λ)𝜈𝜆\nu(\lambda) satisfies

0<ν(0)< and limλ0λν(λ)ν(λ)=0.0𝜈0 and subscript𝜆0𝜆superscript𝜈𝜆𝜈𝜆00<\nu(0)<\infty\text{ and }\lim_{\lambda\to 0}\lambda\frac{\nu^{\prime}(\lambda)}{\nu(\lambda)}=0.

[Assumption A.3] By Tauberian theorem, we have, in (L.1) and (L.2),

limλ(2π)p/2π¯(λ)(λ/2)(p/2αβ1)Γ(p/2αβ1)=1limλ2(2π)p/2π¯(λ)(λ/2)(p/2αβ)Γ(p/2αβ)=1,subscript𝜆superscript2𝜋𝑝2¯𝜋𝜆superscript𝜆2𝑝2𝛼𝛽1Γ𝑝2𝛼𝛽11subscript𝜆2superscript2𝜋𝑝2superscript¯𝜋𝜆superscript𝜆2𝑝2𝛼𝛽Γ𝑝2𝛼𝛽1\begin{split}\lim_{\lambda\to\infty}\frac{(2\pi)^{p/2}\bar{\pi}(\lambda)}{(\lambda/2)^{-(p/2-\alpha-\beta-1)}\Gamma(p/2-\alpha-\beta-1)}=1\\ \lim_{\lambda\to\infty}\frac{-2(2\pi)^{p/2}\bar{\pi}^{\prime}(\lambda)}{(\lambda/2)^{-(p/2-\alpha-\beta)}\Gamma(p/2-\alpha-\beta)}=1,\end{split} (L.5)

which implies that

limλλπ¯(λ)π¯(λ)=p2+α+β+1 and limλλπ(λ)π(λ)=α+β.subscript𝜆𝜆superscript¯𝜋𝜆¯𝜋𝜆𝑝2𝛼𝛽1 and subscript𝜆𝜆superscript𝜋𝜆𝜋𝜆𝛼𝛽\lim_{\lambda\to\infty}\lambda\frac{\bar{\pi}^{\prime}(\lambda)}{\bar{\pi}(\lambda)}=-\frac{p}{2}+\alpha+\beta+1\text{ and }\lim_{\lambda\to\infty}\lambda\frac{\pi^{\prime}(\lambda)}{\pi(\lambda)}=\alpha+\beta. (L.6)

Hence when 1α+β<01𝛼𝛽0-1\leq\alpha+\beta<0, Assumption A.3A.3.1 is satisfied.

When α+β=0𝛼𝛽0\alpha+\beta=0, note

(ξb)α(1+ξ)β=(11+b1+ξ)αsuperscript𝜉𝑏𝛼superscript1𝜉𝛽superscript11𝑏1𝜉𝛼(\xi-b)^{\alpha}(1+\xi)^{\beta}=\left(1-\frac{1+b}{1+\xi}\right)^{\alpha}

and

limξξ{(11+b1+ξ)α1}=α(1+b).subscript𝜉𝜉superscript11𝑏1𝜉𝛼1𝛼1𝑏\lim_{\xi\to\infty}\xi\left\{\left(1-\frac{1+b}{1+\xi}\right)^{\alpha}-1\right\}=-\alpha(1+b).

Then

limξ(2π)p/2π¯(λ)(λ/2)(p/21)Γ(p/21)(λ/2)p/2Γ(p/2)=α(1+b)subscript𝜉superscript2𝜋𝑝2¯𝜋𝜆superscript𝜆2𝑝21Γ𝑝21superscript𝜆2𝑝2Γ𝑝2𝛼1𝑏\lim_{\xi\to\infty}\frac{(2\pi)^{p/2}\bar{\pi}(\lambda)-(\lambda/2)^{-(p/2-1)}\Gamma(p/2-1)}{(\lambda/2)^{-p/2}\Gamma(p/2)}=-\alpha(1+b)

and

limξ2(2π)p/2π¯(λ)(λ/2)p/2Γ(p/2)(λ/2)p/21Γ(p/2+1)=α(1+b).subscript𝜉2superscript2𝜋𝑝2superscript¯𝜋𝜆superscript𝜆2𝑝2Γ𝑝2superscript𝜆2𝑝21Γ𝑝21𝛼1𝑏\lim_{\xi\to\infty}\frac{-2(2\pi)^{p/2}\bar{\pi}^{\prime}(\lambda)-(\lambda/2)^{-p/2}\Gamma(p/2)}{(\lambda/2)^{-p/2-1}\Gamma(p/2+1)}=-\alpha(1+b).

Hence we get

limξλ(λπ¯(λ)π¯(λ)+p21)=limξλ2π(λ)π(λ)=2(p2)α(b+1),subscript𝜉𝜆𝜆superscript¯𝜋𝜆¯𝜋𝜆𝑝21subscript𝜉superscript𝜆2superscript𝜋𝜆𝜋𝜆2𝑝2𝛼𝑏1\lim_{\xi\to\infty}\lambda\left(\lambda\frac{\bar{\pi}^{\prime}(\lambda)}{\bar{\pi}(\lambda)}+\frac{p}{2}-1\right)=\lim_{\xi\to\infty}\lambda^{2}\frac{\pi^{\prime}(\lambda)}{\pi(\lambda)}=2(p-2)\alpha(b+1),

which satisfies Assumption A.3(A.3.2)A.3.2.2. Hence Assumption A.3 is satisfied by 1α+β01𝛼𝛽0-1\leq\alpha+\beta\leq 0. ∎

Appendix M Proof of Admissibility of X𝑋X for p=1,2𝑝12p=1,2

In the Gaussian case, XNp(θ,η1I)similar-to𝑋subscript𝑁𝑝𝜃superscript𝜂1𝐼X\sim N_{p}(\theta,\eta^{-1}I) and ηU2χn2similar-to𝜂superscriptnorm𝑈2subscriptsuperscript𝜒2𝑛\eta\|U\|^{2}\sim\chi^{2}_{n}, Kubokawa in his unpublished lecture note written in Japanese, showed that when p=1,2𝑝12p=1,2, the estimator X𝑋X is admissible among all estimators. Here we generalize it for our general situation with the underlying density f𝑓f given by (1.1). For a general prior g(θ,η)𝑔𝜃𝜂g(\theta,\eta), we have

δg(x,u)=p0θηη(p+n)/2f(η{xθ2+u2})g(θ,η)dθdηp0ηη(p+n)/2f(η{xθ2+u2})g(θ,η)dθdη=x+p0(θx)ηη(p+n)/2f(η{xθ2+u2})g(θ,η)dθdηp0η(p+n)/2+1f(η{xθ2+u2})g(θ,η)dθdη=xp0η(p+n)/2θF(η{xθ2+u2})g(θ,η)dθdηp0η(p+n)/2+1f(η{xθ2+u2})g(θ,η)dθdη=x+p0η(p+n)/2F(η{xθ2+u2})θg(θ,η)dθdηp0η(p+n)/2+1f(η{xθ2+u2})g(θ,η)dθdη,subscript𝛿𝑔𝑥𝑢subscriptsuperscript𝑝superscriptsubscript0𝜃𝜂superscript𝜂𝑝𝑛2𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂subscriptsuperscript𝑝superscriptsubscript0𝜂superscript𝜂𝑝𝑛2𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂𝑥subscriptsuperscript𝑝superscriptsubscript0𝜃𝑥𝜂superscript𝜂𝑝𝑛2𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛21𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂𝑥subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛2subscript𝜃𝐹𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛21𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂𝑥subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛2𝐹𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2subscript𝜃𝑔𝜃𝜂differential-d𝜃differential-d𝜂subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛21𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2𝑔𝜃𝜂differential-d𝜃differential-d𝜂\begin{split}\delta_{g}(x,u)&=\frac{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\theta\eta\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}\\ &=x+\frac{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}(\theta-x)\eta\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2+1}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}\\ &=x-\frac{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2}\nabla_{\theta}F(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2+1}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}\\ &=x+\frac{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2}F(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})\nabla_{\theta}g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2+1}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta},\end{split}

where F(t)=(1/2)tf(s)ds𝐹𝑡12superscriptsubscript𝑡𝑓𝑠differential-d𝑠F(t)=(1/2)\int_{t}^{\infty}f(s)\mathrm{d}s and the last equality follows from an integration by parts. Hence the estimator X𝑋X is the generalized Bayes estimator with respect to any improper prior which does not depend on θ𝜃\theta, say g(θ,η)=π(η)𝑔𝜃𝜂𝜋𝜂g(\theta,\eta)=\pi(\eta). Further let

gi(θ,η)=hi2(ηθ2)π(η)subscript𝑔𝑖𝜃𝜂subscriptsuperscript2𝑖𝜂superscriptnorm𝜃2𝜋𝜂\displaystyle g_{i}(\theta,\eta)=h^{2}_{i}(\eta\|\theta\|^{2})\pi(\eta)

where hisubscript𝑖h_{i} is given by (4.5). Clearly gi(θ,η)subscript𝑔𝑖𝜃𝜂g_{i}(\theta,\eta) approaches π(η)𝜋𝜂\pi(\eta) as i𝑖i\to\infty. Also gi(θ,η)subscript𝑔𝑖𝜃𝜂g_{i}(\theta,\eta) for any fixed i𝑖i is integrable under the condition

0ηp/2π(η)dη<superscriptsubscript0superscript𝜂𝑝2𝜋𝜂differential-d𝜂\int_{0}^{\infty}\eta^{-p/2}\pi(\eta)\mathrm{d}\eta<\infty (M.1)

since

p0gi(θ,η)dθdηsubscriptsuperscript𝑝superscriptsubscript0subscript𝑔𝑖𝜃𝜂differential-d𝜃differential-d𝜂\displaystyle\int_{\mathbb{R}^{p}}\int_{0}^{\infty}g_{i}(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta =p0ηp/2hi2(ηθ2)ηp/2π(η)dθdηabsentsubscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝2subscriptsuperscript2𝑖𝜂superscriptnorm𝜃2superscript𝜂𝑝2𝜋𝜂differential-d𝜃differential-d𝜂\displaystyle=\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{p/2}h^{2}_{i}(\eta\|\theta\|^{2})\eta^{-p/2}\pi(\eta)\mathrm{d}\theta\mathrm{d}\eta
=cp0λp/21hi2(λ)dλ0ηp/2π(η)dη,absentsubscript𝑐𝑝superscriptsubscript0superscript𝜆𝑝21subscriptsuperscript2𝑖𝜆differential-d𝜆superscriptsubscript0superscript𝜂𝑝2𝜋𝜂differential-d𝜂\displaystyle=c_{p}\int_{0}^{\infty}\lambda^{p/2-1}h^{2}_{i}(\lambda)\mathrm{d}\lambda\int_{0}^{\infty}\eta^{-p/2}\pi(\eta)\mathrm{d}\eta,

where, by Lemma E.2 of Appendix E, 0λp/21hi2(λ)dλ<superscriptsubscript0superscript𝜆𝑝21subscriptsuperscript2𝑖𝜆differential-d𝜆\int_{0}^{\infty}\lambda^{p/2-1}h^{2}_{i}(\lambda)\mathrm{d}\lambda<\infty for p=1,2𝑝12p=1,2.

Theorem M.1.

Assume Assumptions F.1, F.2 and F.3F.3.1 on f𝑓f. Then the estimator X𝑋X is admissible for p=1,2𝑝12p=1,2.

Proof.

Let δgisubscript𝛿𝑔𝑖\delta_{gi} be the proper Bayes estimator with respect to gi(θ,η)subscript𝑔𝑖𝜃𝜂g_{i}(\theta,\eta). Then the Bayes risk difference of x𝑥x and δgisubscript𝛿𝑔𝑖\delta_{gi} with respect to gi(θ,η)subscript𝑔𝑖𝜃𝜂g_{i}(\theta,\eta) is

Δi=p0{R(θ,η,X)R(θ,η,δgi)}gi(θ,η)dθdη=pnp0η{xθ2δgiθ2}×η(p+n)/2f(η{xθ2+u2})gi(θ,η)dxdudθdη=pnp0δgix2×η(p+n)/2+1f(η{xθ2+u2})gi(θ,η)dxdudθdη=pnp0η(p+n)/2F(η{xθ2+u2})θgi(θ,η)dθdη2p0η(p+n)/2+1f(η{xθ2+u2})gi(θ,η)dθdηdxdu.subscriptΔ𝑖subscriptsuperscript𝑝superscriptsubscript0𝑅𝜃𝜂𝑋𝑅𝜃𝜂subscript𝛿𝑔𝑖subscript𝑔𝑖𝜃𝜂differential-d𝜃differential-d𝜂subscriptsuperscript𝑝subscriptsuperscript𝑛subscriptsuperscript𝑝superscriptsubscript0𝜂superscriptdelimited-∥∥𝑥𝜃2superscriptdelimited-∥∥subscript𝛿𝑔𝑖𝜃2superscript𝜂𝑝𝑛2𝑓𝜂superscriptdelimited-∥∥𝑥𝜃2superscriptdelimited-∥∥𝑢2subscript𝑔𝑖𝜃𝜂d𝑥d𝑢d𝜃d𝜂subscriptsuperscript𝑝subscriptsuperscript𝑛subscriptsuperscript𝑝superscriptsubscript0superscriptdelimited-∥∥subscript𝛿𝑔𝑖𝑥2superscript𝜂𝑝𝑛21𝑓𝜂superscriptdelimited-∥∥𝑥𝜃2superscriptdelimited-∥∥𝑢2subscript𝑔𝑖𝜃𝜂d𝑥d𝑢d𝜃d𝜂subscriptsuperscript𝑝subscriptsuperscript𝑛superscriptnormsubscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛2𝐹𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2subscript𝜃subscript𝑔𝑖𝜃𝜂differential-d𝜃differential-d𝜂2subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛21𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2subscript𝑔𝑖𝜃𝜂differential-d𝜃differential-d𝜂differential-d𝑥differential-d𝑢\begin{split}\Delta_{i}&=\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\left\{R(\theta,\eta,X)-R(\theta,\eta,\delta_{gi})\right\}g_{i}(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta\\ &=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta\left\{\|x-\theta\|^{2}-\|\delta_{gi}-\theta\|^{2}\right\}\\ &\quad\times\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g_{i}(\theta,\eta)\mathrm{d}x\mathrm{d}u\mathrm{d}\theta\mathrm{d}\eta\\ &=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\|\delta_{gi}-x\|^{2}\\ &\quad\times\eta^{(p+n)/2+1}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g_{i}(\theta,\eta)\mathrm{d}x\mathrm{d}u\mathrm{d}\theta\mathrm{d}\eta\\ &=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{n}}\frac{\left\|\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2}F(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})\nabla_{\theta}g_{i}(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta\right\|^{2}}{\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2+1}f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})g_{i}(\theta,\eta)\mathrm{d}\theta\mathrm{d}\eta}\mathrm{d}x\mathrm{d}u.\end{split}

Note

θgi(θ,η)=4ηθhi(ηθ2)hi(ηθ2).subscript𝜃subscript𝑔𝑖𝜃𝜂4𝜂𝜃subscript𝑖𝜂superscriptnorm𝜃2subscriptsuperscript𝑖𝜂superscriptnorm𝜃2\displaystyle\nabla_{\theta}g_{i}(\theta,\eta)=4\eta\theta h_{i}(\eta\|\theta\|^{2})h^{\prime}_{i}(\eta\|\theta\|^{2}).

Then, by Cauchy-Schwarz inequality, we have

Δipnp0η(p+n)/21F(η{xθ2+u2})2f(η{xθ2+u2})×θgi(θ,η)2gi(θ,η)dxdudθdη=cp+n0t(p+n)/21F2(t)f(t)dtp0θgi(θ,η)2ηgi(θ,η)dθdη=16cp+nA2p0ηp/2ηθ2{hi(ηθ2)}2ηp/2π(η)dθdη=16cpcp+nA20π(η)ηp/2dη0λp/2supi{hi(λ)}2dλsubscriptΔ𝑖subscriptsuperscript𝑝subscriptsuperscript𝑛subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝𝑛21𝐹superscript𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢22𝑓𝜂superscriptnorm𝑥𝜃2superscriptnorm𝑢2superscriptnormsubscript𝜃subscript𝑔𝑖𝜃𝜂2subscript𝑔𝑖𝜃𝜂d𝑥d𝑢d𝜃d𝜂subscript𝑐𝑝𝑛superscriptsubscript0superscript𝑡𝑝𝑛21superscript𝐹2𝑡𝑓𝑡differential-d𝑡subscriptsuperscript𝑝superscriptsubscript0superscriptnormsubscript𝜃subscript𝑔𝑖𝜃𝜂2𝜂subscript𝑔𝑖𝜃𝜂differential-d𝜃differential-d𝜂16subscript𝑐𝑝𝑛subscript𝐴2subscriptsuperscript𝑝superscriptsubscript0superscript𝜂𝑝2𝜂superscriptdelimited-∥∥𝜃2superscriptsubscriptsuperscript𝑖𝜂superscriptdelimited-∥∥𝜃22superscript𝜂𝑝2𝜋𝜂differential-d𝜃differential-d𝜂16subscript𝑐𝑝subscript𝑐𝑝𝑛subscript𝐴2superscriptsubscript0𝜋𝜂superscript𝜂𝑝2differential-d𝜂superscriptsubscript0superscript𝜆𝑝2subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜆2d𝜆\begin{split}\Delta_{i}&\leq\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{(p+n)/2-1}\frac{F(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})^{2}}{f(\eta\{\|x-\theta\|^{2}+\|u\|^{2}\})}\\ &\quad\times\frac{\|\nabla_{\theta}g_{i}(\theta,\eta)\|^{2}}{g_{i}(\theta,\eta)}\mathrm{d}x\mathrm{d}u\mathrm{d}\theta\mathrm{d}\eta\\ &=c_{p+n}\int_{0}^{\infty}t^{(p+n)/2-1}\frac{F^{2}(t)}{f(t)}\mathrm{d}t\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\frac{\|\nabla_{\theta}g_{i}(\theta,\eta)\|^{2}}{\eta g_{i}(\theta,\eta)}\mathrm{d}\theta\mathrm{d}\eta\\ &=16c_{p+n}A_{2}\int_{\mathbb{R}^{p}}\int_{0}^{\infty}\eta^{p/2}\eta\|\theta\|^{2}\left\{h^{\prime}_{i}(\eta\|\theta\|^{2})\right\}^{2}\eta^{-p/2}\pi(\eta)\mathrm{d}\theta\mathrm{d}\eta\\ &=16c_{p}c_{p+n}A_{2}\int_{0}^{\infty}\frac{\pi(\eta)}{\eta^{p/2}}\mathrm{d}\eta\int_{0}^{\infty}\lambda^{p/2}\sup_{i}\left\{h^{\prime}_{i}(\lambda)\right\}^{2}\mathrm{d}\lambda\end{split} (M.2)

where A2=0t(p+n)/21{F2(t)/f(t)}dtsubscript𝐴2superscriptsubscript0superscript𝑡𝑝𝑛21superscript𝐹2𝑡𝑓𝑡differential-d𝑡A_{2}=\int_{0}^{\infty}t^{(p+n)/2-1}\{F^{2}(t)/f(t)\}\mathrm{d}t and it is bounded under Assumptions F.1, F.2 and F.3F.3.1 on f𝑓f, as in Part 11.B of Lemma E.3. Further, by Lemma E.2 of Appendix E, 0λp/2supi{hi(λ)}2dλ<superscriptsubscript0superscript𝜆𝑝2subscriptsupremum𝑖superscriptsubscriptsuperscript𝑖𝜆2d𝜆\int_{0}^{\infty}\lambda^{p/2}\sup_{i}\left\{h^{\prime}_{i}(\lambda)\right\}^{2}\mathrm{d}\lambda<\infty for p=1,2𝑝12p=1,2. Hence, by the dominated convergence theorem, we have Δi0subscriptΔ𝑖0\Delta_{i}\to 0 as i𝑖i\to\infty. By the Blyth sufficient condition, the admissibility of X𝑋X for p=1,2𝑝12p=1,2 follows. ∎

References

  • Blyth (1951) {barticle}[author] \bauthor\bsnmBlyth, \bfnmColin R.\binitsC. R. (\byear1951). \btitleOn minimax statistical decision procedures and their admissibility. \bjournalAnn. Math. Statist. \bvolume22 \bpages22–42. \endbibitem
  • Brewster and Zidek (1974) {barticle}[author] \bauthor\bsnmBrewster, \bfnmJ. F.\binitsJ. F. and \bauthor\bsnmZidek, \bfnmJ. V.\binitsJ. V. (\byear1974). \btitleImproving on equivariant estimators. \bjournalAnn. Statist. \bvolume2 \bpages21–38. \bmrnumber0381098 \endbibitem
  • Brown (1971) {barticle}[author] \bauthor\bsnmBrown, \bfnmL. D.\binitsL. D. (\byear1971). \btitleAdmissible estimators, recurrent diffusions, and insoluble boundary value problems. \bjournalAnn. Math. Statist. \bvolume42 \bpages855–903. \bmrnumber0286209 \endbibitem
  • Brown and Hwang (1982) {bincollection}[author] \bauthor\bsnmBrown, \bfnmLawrence D.\binitsL. D. and \bauthor\bsnmHwang, \bfnmJiunn Tzon\binitsJ. T. (\byear1982). \btitleA unified admissibility proof. In \bbooktitleStatistical decision theory and related topics, III, Vol. 1 (West Lafayette, Ind., 1981) \bpages205–230. \bpublisherAcademic Press, \baddressNew York. \bmrnumber705290 \endbibitem
  • Cellier, Fourdrinier and Robert (1989) {barticle}[author] \bauthor\bsnmCellier, \bfnmDominique\binitsD., \bauthor\bsnmFourdrinier, \bfnmDominique\binitsD. and \bauthor\bsnmRobert, \bfnmChristian\binitsC. (\byear1989). \btitleRobust shrinkage estimators of the location parameter for elliptically symmetric distributions. \bjournalJ. Multivariate Anal. \bvolume29 \bpages39–52. \bdoi10.1016/0047-259X(89)90075-4. \bmrnumber991055 \endbibitem
  • James and Stein (1961) {bincollection}[author] \bauthor\bsnmJames, \bfnmW.\binitsW. and \bauthor\bsnmStein, \bfnmCharles\binitsC. (\byear1961). \btitleEstimation with quadratic loss. In \bbooktitleProc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I \bpages361–379. \bpublisherUniv. California Press, \baddressBerkeley, Calif. \bmrnumber0133191 \endbibitem
  • Kubokawa (2009) {barticle}[author] \bauthor\bsnmKubokawa, \bfnmTatsuya\binitsT. (\byear2009). \btitleIntegral inequality for minimaxity in the Stein problem. \bjournalJ. Japan Statist. Soc. \bvolume39 \bpages155–175. \bmrnumber2722686 \endbibitem
  • Maruyama (2003) {barticle}[author] \bauthor\bsnmMaruyama, \bfnmYuzo\binitsY. (\byear2003). \btitleA robust generalized Bayes estimator improving on the James-Stein estimator for spherically symmetric distributions. \bjournalStatist. Decisions \bvolume21 \bpages69–77. \bmrnumber1985652 \endbibitem
  • Maruyama (2009) {barticle}[author] \bauthor\bsnmMaruyama, \bfnmYuzo\binitsY. (\byear2009). \btitleA Bayes factor with reasonable model selection consistency for ANOVA model. \bjournalArxiv \banumber0906.4329v1 [stat.ME]. \endbibitem
  • Maruyama and Strawderman (2005) {barticle}[author] \bauthor\bsnmMaruyama, \bfnmYuzo\binitsY. and \bauthor\bsnmStrawderman, \bfnmWilliam E.\binitsW. E. (\byear2005). \btitleA new class of generalized Bayes minimax ridge regression estimators. \bjournalAnn. Statist. \bvolume33 \bpages1753–1770. \bmrnumber2166561 \endbibitem
  • Maruyama and Strawderman (2009) {barticle}[author] \bauthor\bsnmMaruyama, \bfnmYuzo\binitsY. and \bauthor\bsnmStrawderman, \bfnmWilliam E.\binitsW. E. (\byear2009). \btitleAn extended class of minimax generalized Bayes estimators of regression coefficients. \bjournalJ. Multivariate Anal. \bvolume100 \bpages2155–2166. \bmrnumber2560360 \endbibitem
  • Maruyama and Strawderman (2017) {barticle}[author] \bauthor\bsnmMaruyama, \bfnmYuzo\binitsY. and \bauthor\bsnmStrawderman, \bfnmWilliam E.\binitsW. E. (\byear2017). \btitleA sharp boundary for SURE-based admissibility for the normal means problem under unknown scale. \bjournalJ. Multivariate Anal. \bpagesaccepted. \endbibitem
  • Stein (1956) {binproceedings}[author] \bauthor\bsnmStein, \bfnmCharles\binitsC. (\byear1956). \btitleInadmissibility of the usual estimator for the mean of a multivariate normal distribution. In \bbooktitleProceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. I \bpages197–206. \bpublisherUniversity of California Press, \baddressBerkeley and Los Angeles. \bmrnumber0084922 \endbibitem
  • Strawderman (1971) {barticle}[author] \bauthor\bsnmStrawderman, \bfnmWilliam E.\binitsW. E. (\byear1971). \btitleProper Bayes minimax estimators of the multivariate normal mean. \bjournalAnn. Math. Statist. \bvolume42 \bpages385–388. \bmrnumber0397939 \endbibitem
  • Weil (1940) {bbook}[author] \bauthor\bsnmWeil, \bfnmAndré\binitsA. (\byear1940). \btitleL’intégration dans les groupes topologiques et ses applications. \bseriesActual. Sci. Ind., no. 869. \bpublisherHermann et Cie., Paris \bnote[This book has been republished by the author at Princeton, N. J., 1941.]. \bmrnumber0005741 \endbibitem
  • Wells and Zhou (2008) {barticle}[author] \bauthor\bsnmWells, \bfnmMartin T.\binitsM. T. and \bauthor\bsnmZhou, \bfnmGongfu\binitsG. (\byear2008). \btitleGeneralized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance. \bjournalJ. Multivariate Anal. \bvolume99 \bpages2208–2220. \bmrnumber2463384 \endbibitem